MATH 216: FOUNDATIONS OF ALGEBRAIC...

MATH 216: FOUNDATIONS OF ALGEBRAIC

GEOMETRY

August 26, 2010.

c! 2010 by Ravi Vakil.

Contents

Chapter 1. Introduction 91.1. Goals 91.2. Background and conventions 11

Part I. Preliminaries 13

Chapter 2. Some category theory 152.1. Motivation 152.2. Categories and functors 172.3. Universal properties determine an object up to unique isomorphism 222.4. Limits and colimits 282.5. Adjoints 322.6. Kernels, cokernels, and exact sequences: A brief introduction to abelian categories 342.7. ! Spectral sequences 43

Chapter 3. Sheaves 553.1. Motivating example: The sheaf of differentiable functions. 553.2. Definition of sheaf and presheaf 573.3. Morphisms of presheaves and sheaves 623.4. Properties determined at the level of stalks, and sheafification 653.5. Sheaves of abelian groups, and OX-modules, form abelian categories 683.6. The inverse image sheaf 703.7. Recovering sheaves from a “sheaf on a base” 72

Part II. Schemes 75

Chapter 4. Toward affine schemes: the underlying set, and the underlying topological space 774.1. Toward schemes 774.2. The underlying set of affine schemes 794.3. Visualizing schemes I: generic points 884.4. The Zariski topology: The underlying topological space of an affine scheme 894.5. A base of the Zariski topology on Spec A: Distinguished open sets 924.6. Topological definitions 934.7. The function I(·), taking subsets of Spec A to ideals of A 98

Chapter 5. The structure sheaf of an affine scheme, and the definition of schemes in general1015.1. The structure sheaf of an affine scheme 1015.2. Visualizing schemes II: nilpotents 1035.3. Definition of schemes 1055.4. Three examples 108

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5.5. Projective schemes 114

Chapter 6. Some properties of schemes 1216.1. Topological properties 1216.2. Reducedness and integrality 1226.3. Properties of schemes that can be checked “affine-locally” 1246.4. Normality and factoriality 1286.5. ! Associated points of (locally Noetherian) schemes, and drawing fuzzy pictures131

Part III. Morphisms of schemes 137

Chapter 7. Morphisms of schemes 1397.1. Introduction 1397.2. Morphisms of ringed spaces 1407.3. From local-ringed spaces to morphisms of schemes 1427.4. Maps of graded rings and maps of projective schemes 1467.5. Rational maps from integral schemes 1487.6. !! Representable functors and group schemes 1537.7. !! The Grassmannian (initial construction) 156

Chapter 8. Useful classes of morphisms of schemes 1598.1. Open immersions 1598.2. Algebraic interlude: Integral morphisms, the Going-Up theorem, and Nakayama’s lemma1608.3. Finiteness conditions on morphisms 1648.4. Images of morphisms: Chevalley’s theorem and elimination theory 171

Chapter 9. Closed immersions and related notions 1779.1. Closed immersions and closed subschemes 1779.2. Closed immersions of projective schemes, and more projective geometry1819.3. Constructions related to “smallest closed subschemes”: scheme-theoretic image, scheme-theoretic closur

Chapter 10. Fibered products of schemes 19110.1. They exist 19110.2. Computing fibered products in practice 19610.3. Pulling back families and fibers of morphisms 19910.4. Properties preserved by base change 20210.5. Products of projective schemes: The Segre embedding 20410.6. Normalization 206

Chapter 11. Separated and proper morphisms, and (finally!) varieties 21311.1. Separated morphisms (and quasiseparatedness done properly) 21311.2. Rational maps to separated schemes 22311.3. Proper morphisms 225

Part IV. Harder properties of schemes 229

Chapter 12. Dimension 23112.1. Dimension and codimension 23112.2. Dimension, transcendence degree, and the Noether normalization lemma23412.3. Fun in codimension one: Krull’s Principal Ideal Theorem and Algebraic Hartogs’ Lemma23812.4. !! Proof of Krull’s Principal Ideal Theorem 12.3.3 243

Chapter 13. Nonsingularity (“smoothness”) of Noetherian schemes 24713.1. The Zariski tangent space 24713.2. The local dimension is at most the dimension of the tangent space, and nonsingularity25113.3. Discrete valuation rings: Dimension 1 Noetherian regular local rings25513.4. Valuative criteria for separatedness and properness 26113.5. ! Completions 264

Part V. Quasicoherent sheaves 267

Chapter 14. Quasicoherent and coherent sheaves 26914.1. Vector bundles and locally free sheaves 26914.2. Quasicoherent sheaves 27314.3. Characterizing quasicoherence using the distinguished affine base 27514.4. Quasicoherent sheaves form an abelian category 27814.5. Module-like constructions 28014.6. Finiteness conditions on quasicoherent sheaves: finite type quasicoherent sheaves, and coherent sheaves14.7. Pleasant properties of finite type and coherent sheaves 28514.8. !! Coherent modules over non-Noetherian rings 288

Chapter 15. Invertible sheaves (line bundles) and divisors 29115.1. Some line bundles on projective space 29115.2. Invertible sheaves and Weil divisors 29315.3. ! Effective Cartier divisors “=” invertible ideal sheaves 300

Chapter 16. Quasicoherent sheaves on projective A-schemes and graded modules30316.1. The quasicoherent sheaf corresponding to a graded module 30316.2. Invertible sheaves (line bundles) on projective A-schemes 30416.3. (Finite) global generation of quasicoherent sheaves, and Serre’s Theorem30516.4. !! Every quasicoherent sheaf on a projective A-scheme arises from a graded module307

Chapter 17. Pushforwards and pullbacks of quasicoherent sheaves 31117.1. Introduction 31117.2. Pushforwards of quasicoherent sheaves 31117.3. Pullbacks of quasicoherent sheaves 31217.4. Invertible sheaves and maps to projective schemes 31617.5. Extending maps to projective schemes over smooth codimension one points: the Curve-to-projective17.6. ! The Grassmannian as a moduli space 322

Chapter 18. Relative Spec and Proj, and projective morphisms 32518.1. Relative Spec of a (quasicoherent) sheaf of algebras 32518.2. Relative Proj of a sheaf of graded algebras 32718.3. Projective morphisms 32918.4. Applications to curves 331

Chapter 19. ! Blowing up a scheme along a closed subscheme 33719.1. Motivating example: blowing up the origin in the plane 33719.2. Blowing up, by universal property 33819.3. The blow-up exists, and is projective 34219.4. Explicit computations 345

Chapter 20. Cech cohomology of quasicoherent sheaves 347

20.1. (Desired) properties of cohomology 34720.2. Definitions and proofs of key properties 35120.3. Cohomology of line bundles on projective space 35620.4. Applications of cohomology: Riemann-Roch, degrees of lines bundles and coherent sheaves, arithmetic20.5. Another application: Hilbert polynomials, genus, and Hilbert functions36120.6. Yet another application: Intersection theory on a nonsingular projective surface36620.7. Higher direct image sheaves 36820.8. ! Higher pushforwards of coherent sheaves under proper morphisms are coherent, and Chow’s lemma

Chapter 21. Curves 37521.1. A criterion for a morphism to be a closed immersion 37521.2. A series of crucial observations 37821.3. Curves of genus 0 38021.4. Hyperelliptic curves 38121.5. Curves of genus 2 38521.6. Curves of genus 3 38621.7. Curves of genus 4 and 5 38721.8. Curves of genus 1 39021.9. Classical geometry involving elliptic curves 39821.10. Counterexamples and pathologies from elliptic curves 398

Chapter 22. Differentials 40122.1. Motivation and game plan 40122.2. The affine case: three definitions 40222.3. Examples 41322.4. Differentials, nonsingularity, and k-smoothness 41722.5. The Riemann-Hurwitz Formula 42122.6. Bertini’s theorem 42422.7. The conormal exact sequence for nonsingular varieties, and useful applications428

Part VI. More 431

Chapter 23. Derived functors 43323.1. The Tor functors 43323.2. Derived functors in general 43623.3. Fun with spectral sequences and derived functors 43823.4. ! Cohomology of O-modules 44023.5. ! Cech cohomology and derived functor cohomology agree 441

Chapter 24. Flatness 44724.1. Introduction 44724.2. Easy facts 44824.3. Flatness through Tor 45324.4. Ideal-theoretic criteria for flatness 45424.5. Flatness implies constant Euler characteristic 45824.6. Statements and applications of cohomology and base change theorems46224.7. ! Universally computing cohomology of flat sheaves using a complex of vector bundles, and proofs of24.8. !! Fancy flatness facts 470

Chapter 25. Smooth, etale, unramified 473

25.1. Some motivation 47325.2. Definitions and easy consequences 47425.3. Harder facts: Left-exactness of the relative cotangent and conormal sequences in the presence of smoothness25.4. Generic smoothness results 47825.5. !! Formally unramified, smooth, and etale 481

Chapter 26. Proof of Serre duality 48326.1. Introduction 48326.2. Serre duality holds for projective space 48426.3. Ext groups and Ext sheaves 48526.4. Serre duality for projective k-schemes 48826.5. Strong Serre duality for particularly nice projective k-schemes 48926.6. The adjunction formula for the dualizing sheaf, and!X = det"X for smooth X491

Bibliography 493

Index 495

CHAPTER 1

Introduction

I can illustrate the .... approach with the ... image of a nut to be opened. The firstanalogy that came to my mind is of immersing the nut in some softening liquid, and whynot simply water? From time to time you rub so the liquid penetrates better, and otherwiseyou let time pass. The shell becomes more flexible through weeks and months — when thetime is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago. The unknown thing to be knownappeared to me as some stretch of earth or hard marl, resisting penetration ... the seaadvances insensibly in silence, nothing seems to happen, nothing moves, the water is sofar off you hardly hear it ... yet finally it surrounds the resistant substance.

— Grothendieck, Recoltes et Semailles p. 552-3, translation by Colin McLarty

1.1 Goals

These are an updated version of notes accompanying a hard year-long classtaught at Stanford in 2009-2010. I am currently editing them and adding a fewmore sections, and I hope a reasonably complete (if somewhat rough) version overthe 2010-11 academic year at the site http://math216.wordpress.com/.

In any class, choices must be made as to what the course is about, and who itis for — there is a finite amount of time, and any addition of material or explana-tion or philosophy requires a corresponding subtraction. So these notes are highlyinappropriate for most people and most classes. Here are my goals. (I do not claimthat these goals are achieved; but they motivate the choices made.)

These notes currently have a very particular audience in mind: Stanford Ph.D.students, postdocs and faculty in a variety of fields, who may want to use alge-braic geometry in a sophisticated way. This includes algebraic and arithmetic ge-ometers, but also topologists, number theorists, symplectic geometers, and others.

The notes deal purely with the algebraic side of the subject, and completelyneglect analytic aspects.

They assume little prior background (see §1.2), and indeed most students havelittle prior background. Readers with less background will necessarily have towork harder. It would be great if the reader had seen varieties before, but manystudents haven’t, and the course does not assume it — and similarly for categorytheory, homological algebra, more advanced commutative algebra, differential ge-ometry, . . . . Surprisingly often, what we need can be developed quickly fromscratch. The cost is that the course is much denser; the benefit is that more peoplecan follow it; they don’t reach a point where they get thrown. (On the other hand,people who already have some familiarity with algebraic geometry, but want to

9

10 Math 216: Foundations of Algebraic Geometry

understand the foundations more completely should not be bored, and will focuson more subtle issues.)

The notes seek to cover everything that one should see in a first course in thesubject, including theorems, proofs, and examples.

They seek to be complete, and not leave important results as black boxespulled from other references.

There are lots of exercises. I have found that unless I have some problems Ican think through, ideas don’t get fixed in my mind. Some are trivial — that’sokay, and even desirable. As few necessary ones as possible should be hard, butthe reader should have the background to deal with them — they are not just anexcuse to push material out of the text.

There are optional (starred !) sections of topics worth knowing on a secondor third (but not first) reading. You should not read double-starred sections (!!)unless you really really want to, but you should be aware of their existence.

The notes are intended to be readable, although certainly not easy reading.In short, after a year of hard work, students should have a broad familiarity

with the foundations of the subject, and be ready to attend seminars, and learnmore advanced material. They should not just have a vague intuitive understand-ing of the ideas of the subject; they should know interesting examples, know whythey are interesting, and be able to prove interesting facts about them.

I have greatly enjoyed thinking through these notes, and teaching the corre-sponding classes, in a way I did not expect. I have had the chance to think throughthe structure of algebraic geometry from scratch, not blindly accepting the choicesmade by others. (Why do we need this notion? Aha, this forces us to consider thisother notion earlier, and now I see why this third notion is so relevant...) I haverepeatedly realized that ideas developed in Paris in the 1960’s are simpler than Iinitially believed, once they are suitably digested.

1.1.1. Implications. We will work with as much generality as we need for mostreaders, and no more. In particular, we try to have hypotheses that are as generalas possible without making proofs harder. The right hypotheses can make a proofeasier, not harder, because one can remember how they get used. As an inflamma-tory example, the notion of quasiseparated comes up early and often. The cost isthat one extra word has to be remembered, on top of an overwhelming numberof other words. But once that is done, it is not hard to remember that essentiallyevery scheme anyone cares about is quasiseparated. Furthermore, whenever thehypotheses “quasicompact and quasiseparated” turn up, the reader will likely im-mediately see a key idea of the proof.

Similarly, there is no need to work over an algebraically closed field, or even afield. Geometers needn’t be afraid of arithmetic examples or of algebraic examples;a central insight of algebraic geometry is that the same formalism applies withoutchange.

1.1.2. Costs. Choosing these priorities requires that others be shortchanged, andit is best to be up front about these. Because of our goal is to be comprehensive,and to understand everything one should know after a first course, it will neces-sarily take longer to get to interesting sample applications. You may be misledinto thinking that one has to work this hard to get to these applications — it is nottrue!

August 26, 2010. 11

1.2 Background and conventions

All rings are assumed to be commutative unless explicitly stated otherwise.All rings are assumed to contain a unit, denoted 1. Maps of rings must send 1 to1. We don’t require that 0 != 1; in other words, the “0-ring” (with one element) isa ring. (There is a ring map from any ring to the 0-ring; the 0-ring only maps toitself. The 0-ring is the final object in the category of rings.) We accept the axiomof choice. In particular, any proper ideal in a ring is contained in a maximal ideal.(The axiom of choice also arises in the argument that the category of A-moduleshas enough injectives, see Exercise 23.2.E.)

The reader should be familiar with some basic notions in commutative ringtheory, in particular the notion of ideals (including prime and maximal ideals)and localization. For example, the reader should be able to show that if S is amultiplicative set of a ring A (which we assume to contain 1), then the primes ofS!1A are in natural bijection with those primes of A not meeting S (§4.2.6). Tensorproducts and exact sequences of A-modules will be important. We will use thenotation (A,m) or (A,m, k) for local rings — A is the ring, m its maximal ideal,and k = A/m its residue field. We will use (in Proposition 14.7.1) the structuretheorem for finitely generated modules over a principal ideal domain A: any suchmodule can be written as the direct sum of principal modules A/(a).

We will not concern ourselves with subtle foundational issues (set-theoreticissues involving universes, etc.). It is true that some people should be carefulabout these issues. But is that really how you want to spend your life? (If so, agood start is [KS, §1.1].)

1.2.1. Further background. It may be helpful to have books on other subjectshandy that you can dip into for specific facts, rather than reading them in ad-vance. In commutative algebra, Eisenbud [E] is good for this. Other popularchoices are Atiyah-Macdonald [AM] and Matsumura [M-CRT]. For homologicalalgebra, Weibel [W] is simultaneously detailed and readable.

Background from other parts of mathematics (topology, geometry, complexanalysis) will of course be helpful for developing intuition.

Finally, it may help to keep the following quote in mind.

Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive,and very abstract, with adherents who are secretly plotting to take over all the rest ofmathematics. In one respect this last point is accurate.

— David Mumford

Part I

Preliminaries

CHAPTER 2

Some category theory

That which does not kill me, makes me stronger. — Nietzsche

2.1 Motivation

Before we get to any interesting geometry, we need to develop a languageto discuss things cleanly and effectively. This is best done in the language ofcategories. There is not much to know about categories to get started; it is justa very useful language. Like all mathematical languages, category theory comeswith an embedded logic, which allows us to abstract intuitions in settings we knowwell to far more general situations.

Our motivation is as follows. We will be creating some new mathematicalobjects (such as schemes, and certain kinds of sheaves), and we expect them toact like objects we have seen before. We could try to nail down precisely whatwe mean by “act like”, and what minimal set of things we have to check in orderto verify that they act the way we expect. Fortunately, we don’t have to — otherpeople have done this before us, by defining key notions, such as abelian categories,which behave like modules over a ring.

Our general approach will be as follows. I will try to tell what you need toknow, and no more. (This I promise: if I use the word “topoi”, you can shoot me.) Iwill begin by telling you things you already know, and describing what is essentialabout the examples, in a way that we can abstract a more general definition. Wewill then see this definition in less familiar settings, and get comfortable with usingit to solve problems and prove theorems.

For example, we will define the notion of product of schemes. We could justgive a definition of product, but then you should want to know why this precisedefinition deserves the name of “product”. As a motivation, we revisit the notionof product in a situation we know well: (the category of) sets. One way to definethe product of sets U and V is as the set of ordered pairs {(u, v) : u " U, v " V}.But someone from a different mathematical culture might reasonably define it asthe set of symbols {

uv : u " U, v " V}. These notions are “obviously the same”.

Better: there is “an obvious bijection between the two”.This can be made precise by giving a better definition of product, in terms of a

universal property. Given two sets M and N, a product is a set P, along with mapsµ : P ! M and # : P ! N, such that for any set P ! with maps µ ! : P ! ! M and

15


# ! : P ! ! N, these maps must factor uniquely through P:

(2.1.0.1) P !

"!

!!

! !

""!!!!!!!!!!!!!!!

µ !

##""""""""""""""

P !$$

µ

%%

N

M

Thus a product is a diagram

P! $$

µ

%%

N

M

and not just a set P, although the maps µ and # are often left implicit.This definition agrees with the traditional definition, with one twist: there

isn’t just a single product; but any two products come with a unique isomorphismbetween them. In other words, the product is unique up to unique isomorphism.Here is why: if you have a product

P1!1 $$

µ1

%%

N

M

and I have a product

P2!2 $$

µ2

%%

N

M

then by the universal property of my product (letting (P2, µ2,#2) play the role of(P, µ,#), and (P1, µ1,#1) play the role of (P !, µ !,# !) in (2.1.0.1)), there is a uniquemap f : P1 ! P2 making the appropriate diagram commute (i.e. µ1 = µ2 # f and#1 = #2 # f). Similarly by the universal property of your product, there is a uniquemap g : P2 ! P1 making the appropriate diagram commute. Now consider theuniversal property of my product, this time letting (P2, µ2,#2) play the role of both(P, µ,#) and (P !, µ !,# !) in (2.1.0.1). There is a unique map h : P2 ! P2 such that

P2

h

!!##

##

##

#!2

&&!!!!!!!!!!!!!!!

µ2

##"""""""""""""""

P2 !2

$$

µ2

%%

N

M

commutes. However, I can name two such maps: the identity map idP2, and g # f.

Thus g # f = idP2. Similarly, f # g = idP1

. Thus the maps f and g arising fromthe universal property are bijections. In short, there is a unique bijection between

August 26, 2010. 17

P1 and P2 preserving the “product structure” (the maps to M and N). This givesus the right to name any such product M $ N, since any two such products areuniquely identified.

This definition has the advantage that it works in many circumstances, andonce we define categories, we will soon see that the above argument applies ver-batim in any category to show that products, if they exist, are unique up to uniqueisomorphism. Even if you haven’t seen the definition of category before, you canverify that this agrees with your notion of product in some category that you haveseen before (such as the category of vector spaces, where the maps are taken to belinear maps; or the category of smooth manifolds, where the maps are taken to besmooth maps).

This is handy even in cases that you understand. For example, one way ofdefining the product of two manifolds M and N is to cut them both up into charts,then take products of charts, then glue them together. But if I cut up the manifoldsin one way, and you cut them up in another, how do we know our resulting mani-folds are the “same”? We could wave our hands, or make an annoying argumentabout refining covers, but instead, we should just show that they are “categoricalproducts” and hence canonically the “same” (i.e. isomorphic). We will formalizethis argument in §2.3.

Another set of notions we will abstract are categories that “behave like mod-ules”. We will want to define kernels and cokernels for new notions, and weshould make sure that these notions behave the way we expect them to. Thisleads us to the definition of abelian categories, first defined by Grothendieck in hisTohoku paper [Gr].

In this chapter, we’ll give an informal introduction to these and related notions,in the hope of giving just enough familiarity to comfortably use them in practice.

2.2 Categories and functors

We begin with an informal definition of categories and functors.

2.2.1. Categories.A category consists of a collection of objects, and for each pair of objects, a set

of maps, or morphisms (or arrows or maps), between them. The collection of ob-jects of a category C are often denoted obj(C), but we will usually denote the collec-tion also by C. If A,B " C, then the morphisms from A to B are denoted Mor(A,B).A morphism is often written f : A ! B, and A is said to be the source of f, andB the target of f. (Of course, Mor(A,B) is taken to be disjoint from Mor(A !, B !)unless A = A ! and B = B !.)

Morphisms compose as expected: there is a composition Mor(A,B)$Mor(B,C) !Mor(A,C), and if f " Mor(A,B) and g " Mor(B,C), then their composition is de-noted g # f. Composition is associative: if f " Mor(A,B), g " Mor(B,C), andh " Mor(C,D), then h # (g # f) = (h # g) # f. For each object A " C, there is alwaysan identity morphism idA : A ! A, such that when you (left- or right-)composea morphism with the identity, you get the same morphism. More precisely, iff : A ! B is a morphism, then f # idA = f = idB #f. (If you wish, you may check


that “identity morphisms are unique”: there is only on morphism deserving thename idA.)

If we have a category, then we have a notion of isomorphism between twoobjects (a morphism f : A ! B such that there exists some — necessarily unique —morphism g : B ! A, where f#g and g#f are the identity on B and A respectively),and a notion of automorphism of an object (an isomorphism of the object withitself).

2.2.2. Example. The prototypical example to keep in mind is the category of sets,denoted Sets. The objects are sets, and the morphisms are maps of sets. (BecauseRussell’s paradox shows that there is no set of all sets, we did not say earlier thatthere is a set of all objects. But as stated in §1.2, we are deliberately omitting allset-theoretic issues.)

2.2.3. Example. Another good example is the category Veck of vector spaces overa given field k. The objects are k-vector spaces, and the morphisms are lineartransformations. (What are the isomorphisms?)

2.2.A. UNIMPORTANT EXERCISE. A category in which each morphism is an iso-morphism is called a groupoid. (This notion is not important in these notes. Thepoint of this exercise is to give you some practice with categories, by relating themto an object you know well.)(a) A perverse definition of a group is: a groupoid with one object. Make sense ofthis.(b) Describe a groupoid that is not a group.

2.2.B. EXERCISE. If A is an object in a category C, show that the invertible ele-ments of Mor(A,A) form a group (called the automorphism group of A, denotedAut(A)). What are the automorphism groups of the objects in Examples 2.2.2and 2.2.3? Show that two isomorphic objects have isomorphic automorphismgroups. (For readers with a topological background: if X is a topological space,then the fundamental groupoid is the category where the objects are points of x,and the morphisms x ! y are paths from x to y, up to homotopy. Then the auto-morphism group of x0 is the (pointed) fundamental group $1(X, x0). In the casewhere X is connected, and $1(X) is not abelian, this illustrates the fact that fora connected groupoid — whose definition you can guess — the automorphismgroups of the objects are all isomorphic, but not canonically isomorphic.)

2.2.4. Example: abelian groups. The abelian groups, along with group homomor-phisms, form a category Ab.

2.2.5. Important example: modules over a ring. If A is a ring, then the A-modules forma category ModA. (This category has additional structure; it will be the prototypi-cal example of an abelian category, see §2.6.) Taking A = k, we obtain Example 2.2.3;taking A = Z, we obtain Example 2.2.4.

2.2.6. Example: rings. There is a category Rings, where the objects are rings, and themorphisms are morphisms of rings (which send 1 to 1 by our conventions, §1.2).

2.2.7. Example: topological spaces. The topological spaces, along with continuousmaps, form a category Top. The isomorphisms are homeomorphisms.

August 26, 2010. 19

In all of the above examples, the objects of the categories were in obvious wayssets with additional structure. This needn’t be the case, as the next example shows.

2.2.8. Example: partially ordered sets. A partially ordered set, or poset, is a set Salong with a binary relation % on S satisfying:

(i) x % x (reflexivity),(ii) x % y and y % z imply x % z (transitivity), and

(iii) if x % y and y % x then x = y.

A partially ordered set (S,%) can be interpreted as a category whose objects arethe elements of S, and with a single morphism from x to y if and only if x % y (andno morphism otherwise).

A trivial example is (S,%) where x % y if and only if x = y. Another exampleis

(2.2.8.1) •

%%• $$ •

Here there are three objects. The identity morphisms are omitted for convenience,and the two non-identity morphisms are depicted. A third example is

(2.2.8.2) •

%%

$$ •

%%• $$ •

Here the “obvious” morphisms are again omitted: the identity morphisms, andthe morphism from the upper left to the lower right. Similarly,

· · · $$ • $$ • $$ •

depicts a partially ordered set, where again, only the “generating morphisms” aredepicted.

2.2.9. Example: the category of subsets of a set, and the category of open sets in a topo-logical space. If X is a set, then the subsets form a partially ordered set, where theorder is given by inclusion. Similarly, if X is a topological space, then the open setsform a partially ordered set, where the order is given by inclusion.

2.2.10. Example. A subcategory A of a category B has as its objects some of theobjects of B, and some of the morphisms, such that the morphisms of A includethe identity morphisms of the objects of A, and are closed under composition. (Forexample, (2.2.8.1) is in an obvious way a subcategory of (2.2.8.2).)

2.2.11. Functors.A covariant functor F from a category A to a category B, denoted F : A ! B,

is the following data. It is a map of objects F : obj(A) ! obj(B), and for each A1,A2 " A, and morphism m : A1 ! A2, a morphism F(m) : F(A1) ! F(A2) in B. Werequire that F preserves identity morphisms (for A " A, F(idA) = idF(A)), and thatF preserves composition (F(m1 # m2) = F(m1) # F(m2)). (You may wish to verifythat covariant functors send isomorphisms to isomorphisms.)


If F : A ! B and G : B ! C are covariant functors, then we define a functorG # F : A ! C in the obvious way. Composition of functors is associative in anevident sense.

2.2.12. Example: a forgetful functor. Consider the functor from the category ofvector spaces (over a field k) Veck to Sets, that associates to each vector space itsunderlying set. The functor sends a linear transformation to its underlying map ofsets. This is an example of a forgetful functor, where some additional structure isforgotten. Another example of a forgetful functor is ModA ! Ab from A-modulesto abelian groups, remembering only the abelian group structure of the A-module.

2.2.13. Topological examples. Examples of covariant functors include the funda-mental group functor $1, which sends a topological space X with choice of a pointx0 " X to a group $1(X, x0) (what are the objects and morphisms of the source cat-egory?), and the ith homology functor Top ! Ab, which sends a topological spaceX to its ith homology group Hi(X, Z). The covariance corresponds to the fact thata (continuous) morphism of pointed topological spaces f : X ! Y with f(x0) = y0

induces a map of fundamental groups $1(X, x0) ! $1(Y, y0), and similarly forhomology groups.

2.2.14. Example. Suppose A is an object in a category C. Then there is a functor hA :C ! Sets sending B " C to Mor(A,B), and sending f : B1 ! B2 to Mor(A,B1) !Mor(A,B2) described by

[g : A ! B1] &! [f # g : A ! B1 ! B2].

This seemingly silly functor ends up surprisingly being an important concept.

2.2.15. Full and faithful functors. A covariant functor F : A ! B is faithful if forall A,A ! " A, the map MorA(A,A !) ! MorB(F(A), F(A !)) is injective, and full ifit is surjective. A functor that is full and faithful is fully faithful. A subcategoryi : A ! B is a full subcategory if i is full. Thus a subcategory A ! of A is full if andonly if for all A,B " obj(A !), MorA !(A,B) = MorA(A,B).

2.2.16. Definition. A contravariant functor is defined in the same way as a covari-ant functor, except the arrows switch directions: in the above language, F(A1 !A2) is now an arrow from F(A2) to F(A1). (Thus F(m2 # m1)F(m1) # F(m2), notthe other way around.)

It is wise to always state whether a functor is covariant or contravariant. If itis not stated, the functor is often assumed to be covariant.

(Sometimes people describe a contravariant functor C ! D as a covariant func-tor Copp ! D, where Copp is the same category as C except that the arrows go inthe opposite direction. Here Copp is said to be the opposite category to C.)

2.2.17. Linear algebra example. If Veck is the category of k-vector spaces (introducedin Example 2.2.12), then taking duals gives a contravariant functor ·! : Veck !Veck. Indeed, to each linear transformation f : V ! W, we have a dual transforma-tion f! : W! ! V!, and (f # g)! = g! # f!.

2.2.18. Topological example (cf. Example 2.2.13). The the ith cohomology functorHi(·, Z) : Top ! Ab is a contravariant functor.

August 26, 2010. 21

2.2.19. Example. There is a contravariant functor Top ! Rings taking a topologicalspace X to the real-valued continuous functions on X. A morphism of topologicalspaces X ! Y (a continuous map) induces the pullback map from functions on Yto maps on X.

2.2.20. Example (cf. Example 2.2.14). Suppose A is an object of a category C. Thenthere is a contravariant functor hA : C ! Sets sending B " C to Mor(B,A), andsending the morphism f : B1 ! B2 to the morphism Mor(B2, A) ! Mor(B1, A)via

[g : B2 ! A] &! [g # f : B2 ! B1 ! A].

This example initially looks weird and different, but Examples 2.2.17 and 2.2.19may be interpreted as special cases; do you see how? What is A in each case?

2.2.21. ! Natural transformations (and natural isomorphisms) of functors, andequivalences of categories.

(This notion won’t come up in an essential away until at least Chapter 7, soyou shouldn’t read this section until then.) Suppose F and G are two functors fromA to B. A natural transformation of functors F ! G is the data of a morphismma : F(a) ! G(a) for each a " A such that for each f : a ! a ! in A, the diagram

F(a)F(f) $$

ma

%%

F(a !)

ma !

%%G(a)

G(f)$$ G(a !)

commutes. A natural isomorphism of functors is a natural transformation suchthat each ma is an isomorphism. The data of functors F : A ! B and F ! : B ! Asuch that F # F ! is naturally isomorphic to the identity functor IB on B and F ! # F isnaturally isomorphic to IA is said to be an equivalence of categories. This is the“right” notion of isomorphism of categories.

Two examples might make this strange concept more comprehensible. Thedouble dual of a finite-dimensional vector space V is not V , but we learn early tosay that it is canonically isomorphic to V . We make can that precise as follows. Letf.d.Veck be the category of finite-dimensional vector spaces over k. Note that thiscategory contains oodles of vector spaces of each dimension.

2.2.C. EXERCISE. Let ·!! : f.d.Veck ! f.d.Veck be the double dual functor from thecategory of vector spaces over k to itself. Show that ·!! is naturally isomorphicto the identity functor on f.d.Veck. (Without the finite-dimensional hypothesis, weonly get a natural transformation of functors from id to ·!!.)

Let V be the category whose objects are kn for each n (there is one vector spacefor each n), and whose morphisms are linear transformations. This latter space canbe thought of as vector spaces with bases, and the morphisms are honest matrices.There is an obvious functor V ! f.d.Veck, as each kn is a finite-dimensional vectorspace.

2.2.D. EXERCISE. Show that V ! f.d.Veck gives an equivalence of categories,by describing an “inverse” functor. (We are assuming any needed version of the


axiom of choice, §1.2, so feel free to simultaneously choose bases for each vectorspace in f.d.Veck.)

2.2.22. !! Aside for experts. One may show that this definition is equivalent toanother one commonly given: a covariant functor F : A ! B is an equivalence ofcategories if it is fully faithful and every object of B is isomorphic to an object ofthe form F(a) (F is essentially surjective). One can show that such a functor has aquasiinverse, i.e., that there is a functor G : B ! A, which is also an equivalence,and for which there exist natural isomorphisms G(F(A)) != A and F(G(B)) != B.Thus “equivalence of categories” is an equivalence relation. The notion of “equiv-alence of categories” is the right notion of what one thinks of as “isomorphismof categories” (I informally think of it as “essentially the same category”), but thereason for this would take too long to go into here.

2.3 Universal properties determine an object up to uniqueisomorphism

Given some category that we come up with, we often will have ways of pro-ducing new objects from old. In good circumstances, such a definition can bemade using the notion of a universal property. Informally, we wish that there werean object with some property. We first show that if it exists, then it is essentiallyunique, or more precisely, is unique up to unique isomorphism. Then we go aboutconstructing an example of such an object to show existence.

Explicit constructions are sometimes easier to work with than universal prop-erties, but with a little practice, universal properties are useful in proving thingsquickly and slickly. Indeed, when learning the subject, people often find explicitconstruction more appealing, and use them more often in proofs, but as they be-come more experienced, find universal property arguments more elegant and in-sightful.

We have seen one important example of a universal property argument al-ready in §2.1: products. You should go back and verify that our discussion theregives a notion of product in any category, and shows that products, if they exist, areunique up to unique isomorphism.

2.3.1. Localization. A second example of universal property is the notion oflocalization of a ring. We first review a constructive definition, and then reinterpretthe notion in terms of universal property. A multiplicative subset S of a ring Ais a subset closed under multiplication containing 1. We define a ring S!1A. Theelements of S!1A are of the form a/s where a " A and s " S, and define (a1/s1)$(a2/s2) = (a1a2)/(s1s2), and (a1/s1) + (a2/s2) = (s2a1 + s1a2)/(s1s2). We saythat a1/s1 = a2/s2 if for some s " S, s(s2a1 ! s1a2) = 0. (This implies thatS!1A is 0 if 0 " S.) We have a canonical map A ! S!1A given by a &! a/1. IfS = {fn : n " Z#0}, where f " A, we define Af := S!1A.

2.3.A. EXERCISE. Verify that S!1A satisfies the following universal property:S!1A is initial among A-algebras B where every element of S is sent to a unitin B. (Recall: the data of “an A-algebra B” and “a ring map A ! B” the the same.)

August 26, 2010. 23

Warning: sometimes localization is first introduced in the special case whereA is an integral domain. In that case, A "! S!1A, but this isn’t always true, asshown by the following result.

2.3.B. EXERCISE. Show that A ! S!1A is injective if and only if S contains nozero-divisors. (A zero-divisor of a ring A is an element a such that there is a non-zero element b with ab = 0. The other elements of A are called non-zero-divisors.For example, a unit is never a zero-divisor. Counter-intuitively, 0 is a zero-divisorin a ring A if and only if A is not the 0-ring.)

In fact, it is cleaner to define S!1A by this universal property, and to show thatit exists, and to use the universal property to check various properties S!1A has.Let’s get some practice with this by defining localizations of modules by universalproperty. Suppose M is an A-module. Define S!1M as being initial among A-modules N for which s $ · : N ! N is an isomorphism for all s " S.

2.3.C. EXERCISE. Show that if S!1M exists, then (a) there is a natural map M !S!1M (here “natural” is meant informally as “obvious” — you know it if you seeit), and (b) the A-module structure on S!1M extends to an S!1A-module structure.

2.3.D. EXERCISE. Show that S!1M exists, by constructing something satisfyingthe universal property. Hint: define elements of S!1M to be of the form m/swhere m " M and s " S, and m1/s1 = m2/s2 if and only if for some s " S,s(s2m1!s1m2) = 0. Define the additive structure by (m1/s1)+(m2/s2) = (s2m1+s1m2)/(s1s2), and the S!1A-module structure (and hence the A-module structure)is given by (a1/s1) # (m2/s2) = (a1m2)/(s1s2).

2.3.E. EXERCISE. Show that localization commutes with finite products. In otherwords, if M1, . . . , Mn are A-modules, describe an isomorphism S!1(M1 $ · · · $Mn) ! S!1M1 $ · · ·$ S!1Mn.

2.3.2. Tensor products. Another important example of a universal property con-struction is the notion of a tensor product of A-modules

'A : obj(ModA) $ obj(ModA) $$ obj(ModA)

(M,N) $ $$ M 'A N

The subscript A is often suppressed when it is clear from context. The tensor prod-uct is often defined as follows. Suppose you have two A-modules M and N. Thenelements of the tensor product M'AN are finite A-linear combinations of symbolsm ' n (m " M, n " N), subject to relations (m1 + m2) ' n = m1 ' n + m2 ' n,m ' (n1 + n2) = m ' n1 + m ' n2, a(m ' n) = (am) ' n = m ' (an) (wherea " A, m1,m2 " M, n2, n2 " N). More formally, M 'A N is the free A-modulegenreated by M$N, quotiented by the submodule generated by (m1 +m2)'n!m1 ' n ! m2 ' n, m ' (n1 + n2) ! m ' n1 ! m ' n2, a(m ' n) ! (am) ' n, anda(m ' n) ! m ''(an) for a " A, m,m1,m2 " M, n,n1, n2 " N.

If A is a field k, we recover the tensor product of vector spaces.


2.3.F. EXERCISE (IF YOU HAVEN’T SEEN TENSOR PRODUCTS BEFORE). CalculateZ/(10) 'Z Z/(12). (This exercise is intended to give some hands-on practice withtensor products.)

2.3.G. IMPORTANT EXERCISE: RIGHT-EXACTNESS OF · 'A N. Show that · 'A Ngives a covariant functor ModA ! ModA. Show that ·'AN is a right-exact functor,i.e. if

M ! ! M ! M !! ! 0

is an exact sequence of A-modules (which means f : M ! M !! is surjective, andM ! surjects onto the kernel of f; see §2.6), then the induced sequence

M ! 'A N ! M 'A N ! M !! 'A N ! 0

is also exact. (This exercise is repeated in Exercise 2.6.F, but you may get a lotout of doing it now.) (You will be reminded of the definition of right-exactness in§2.6.4.)

The constructive definition ' is a weird definition, and really the “wrong”definition. To motivate a better one: notice that there is a natural A-bilinear mapM $ N ! M 'A N. (If M,N, P " ModA, a map f : M $ N ! P is A-bilinearif f(m1 + n2, n) = f(m1, n) + f(m2, n), f(m,n1 + n2) = f(m,n1) + f(m,n2), andf(am,n) = f(m,an) = af(m,n).) Any A-bilinear map M$N ! C factors throughthe tensor product uniquely: M $ N ! M 'A N ! C. (Think this through!)

We can take this as the definition of the tensor product as follows. It is an A-module T along with an A-bilinear map t : M $ N ! T , such that given anyA-bilinear map t ! : M $ N ! T !, there is a unique A-linear map f : T ! T ! suchthat t ! = f # t.

M $ Nt $$

t !

''%%%%

%%%%

% T

"!f((T !

2.3.H. EXERCISE. Show that (T, t : M $ N ! T) is unique up to unique isomor-phism. Hint: first figure out what “unique up to unique isomorphism” means forsuch pairs. Then follow the analogous argument for the product.

In short: given M and N, there is an A-bilinear map t : M $ N ! M 'A N,unique up to unique isomorphism, defined by the following universal property:for any A-bilinear map t ! : M $ N ! T ! there is a unique A-linear map f : M 'A

N ! T ! such that t ! = f # t.Note that this argument shows uniqueness assuming existence. We need to

still show the existence of such a tensor product. This forces us to do somethingconstructive. Fortunately, we already have:

2.3.I. EXERCISE. Show that the construction of §2.3.2 satisfies the universal prop-erty of tensor product.

The uniqueness of tensor product is our second example of the proof of unique-ness (up to unique isomorphism) by a universal property. If you have never seenthis sort of argument before, then you might think you get it, but you should thinkover it some more. We will be using such arguments repeatedly in the future.

August 26, 2010. 25

The two exercises below are some useful facts about tensor products withwhich you should be familiar. The first exercise deals with localization, so hereis a brief introduction in case you haven’t seen it before.

2.3.J. IMPORTANT EXERCISE. (a) If M is an A-module and A ! B is a morphismof rings, show that B'A M naturally has the structure of a B-module. (In fact thisdescribes a functor ModA ! ModB, but you needn’t show this unless you wantto.)(b) If further A ! C is a morphism of rings, show that B'A C has the structure ofa ring. Hint: multiplication will be given by (b1 ' c1)(b2 ' c2) = (b1b2) ' (c1c2).(Exercise 2.3.Y will interpret this construction as a coproduct.)

2.3.K. IMPORTANT EXERCISE. If S is a multiplicative subset of A and M is an A-module, describe a natural isomorphism (S!1A)'AM != S!1M (as S!1A-modulesand as A-modules).

Here is another exercise involving a universal property.

2.3.3. Definition. An object of a category C is an initial object if it has preciselyone map to every object. It is a final object if it has precisely one map from everyobject. It is a zero object if it is both an initial object and a final object.

2.3.L. EXERCISE. Show that any two initial objects are uniquely isomorphic. Showthat any two final objects are uniquely isomorphic.

This (partially) justifies the phrase “the initial object” rather than “an initialobject”, and similarly for “the final object” and “the zero object”.

2.3.M. EXERCISE. What are the initial and final objects are in Sets, Rings, and Top(if they exist)? How about the two examples of §2.2.9?

2.3.N. ! EXERCISE. Prove Yoneda’s Lemma.

2.3.4. Important Example: Fibered products. (This notion will be essential later.)Suppose we have morphisms f : X ! z and g : Y ! Z (in any category). Thenthe fibered product is an object X $Z Y along with morphisms $X : X $Z Y ! Xand $Y : X $Z Y ! Y, where the two compositions f # $X, g # $Y : X $Z Y ! Zagree, such that given any object W with maps to X and Y (whose compositions toZ agree), these maps factor through some unique W ! X $Z Y:

W

"!

''

))&&

&&

&&

&&

&&

&&

&&

&&

**'''''''''''''''''''

X $Z Y

"X

%%

"Y

$$ Y

g

%%X

f $$ Z

(Warning: the definition of the fibered product depends on f and g, even thoughthey are omitted from the notation X $Z Y.)

By the usual universal property argument, if it exists, it is unique up to uniqueisomorphism. (You should think this through until it is clear to you.) Thus the use


of the phrase “the fibered product” (rather than “a fibered product”) is reasonable,and we should reasonably be allowed to give it the name X $Z Y. We know whatmaps to it are: they are precisely maps to X and maps to Y that agree as maps to Z.

Depending on your religion, the diagram

X $Z Y

"X

%%

"Y

$$ Y

g

%%X

f $$ Z

is called a fibered/pullback/Cartesian diagram/square (six possibilities).The right way to interpret the notion of fibered product is first to think about

what it means in the category of sets.

2.3.O. EXERCISE. Show that in Sets,

X $Z Y = {(x " X, y " Y) : f(x) = g(y)}.

More precisely, show that the right side, equipped with its evident maps to X andY, satisfies the universal property of the fibered product. (This will help you buildintuition for fibered products.)

2.3.P. EXERCISE. If X is a topological space, show that fibered products alwaysexist in the category of open sets of X, by describing what a fibered product is.(Hint: it has a one-word description.)

2.3.Q. EXERCISE. If Z is the final object in a category C, and X, Y " C, show that“X $Z Y = X $ Y”: “the” fibered product over Z is uniquely isomorphic to “the”product. (This is an exercise about unwinding the definition.)

2.3.R. USEFUL EXERCISE: TOWERS OF FIBER DIAGRAMS ARE FIBER DIAGRAMS. Ifthe two squares in the following commutative diagram are fiber diagrams, showthat the “outside rectangle” (involving U, V , Y, and Z) is also a fiber diagram.

U $$

%%

V

%%W $$

%%

X

%%Y $$ Z

2.3.S. EXERCISE. Given X ! Y ! Z, show that there is a natural morphismX $Y X ! X $Z X, assuming that both fibered products exist. (This is trivial onceyou figure out what it is saying. The point of this exercise is to see why it is trivial.)

2.3.T. USEFUL EXERCISE: THE MAGIC DIAGRAM. Suppose we are given mor-phisms X1, X2 ! Y and Y ! Z. Describe the natural morphism X1 $Y X2 !

August 26, 2010. 27

X1 $Z X2. Show that the following diagram is a fibered square.

X1 $Y X2$$

%%

X1 $Z X2

%%Y $$ Y $Z Y

This diagram is surprisingly incredibly useful — so useful that we will call it themagic diagram.

2.3.5. Monomorphisms and epimorphisms.

2.3.6. Definition. A morphism f : X ! Y is a monomorphism if any two mor-phisms g1, g2 : Z ! X such that f # g1 = f # g2 must satisfy g1 = g2. In otherwords, for any other object Z, the natural map Hom(Z,X) ! Hom(Z, Y) is aninjection. This a generalization of an injection of sets. In other words, there is aunique way of filling in the dotted arrow so that the following diagram commutes.

Z

$1

%% ++((

((

((

(

Xf

$$ Y.

Intuitively, it is the categorical version of an injective map, and indeed this notiongeneralizes the familiar notion of injective maps of sets. (The reason we don’t usethe word “injective” is that in some contexts, “injective” will have an intuitivemeaning which may not agree with “monomorphism”. This is also the case with“epimorphism” vs. “surjective”.)

2.3.U. EXERCISE. Show that the composition of two monomorphisms is a monomor-phism.

2.3.V. EXERCISE. Prove a morphism X ! Y is a monomorphism if and only ifthe induced morphism X ! X $Y X is an isomorphism. We may then take thisas the definition of monomorphism. (Monomorphisms aren’t central to futurediscussions, although they will come up again. This exercise is just good practice.)

2.3.W. EXERCISE. Suppose Y ! Z is a monomorphism, and X1, X2 ! Y are twomorphisms. Show that X1$Y X2 and X1$Z X2 are canonically isomorphic. We willuse this later when talking about fibered products. (Hint: for any object V , give anatural bijection between maps from V to the first and maps from V to the second.It is also possible to use the magic diagram, Exercise 2.3.T)

The notion of an epimorphism is “dual” to the definition of monomorphism,where all the arrows are reversed. This concept will not be central for us, althoughit is necessary for the definition of an abelian category. Intuitively, it is the categor-ical version of a surjective map.

2.3.7. Coproducts.


2.3.X. EXERCISE. Define coproduct in a category by reversing all the arrows in thedefinition of product. Show that coproduct for Sets is disjoint union. (This is whywe use the notation

"for disjoint union.)

2.3.Y. EXERCISE. Suppose A ! B,C are two ring morphisms, so in particular Band C are A-modules. Recall (Exercise 2.3.J) that B'A C has a ring structure. Showthat there is a natural morphism B ! B 'A C given by b &! b ' 1. (This is notnecessarily an inclusion, see Exercise 2.3.F.) Similarly, there is a natural morphismC ! B 'A C. Show that this gives a fibered coproduct on rings, i.e. that

B 'A C C,,

B

--

A,,

--

satisfies the universal property of fibered coproduct.

2.3.Z. ! EXERCISE (REPRESENTABLE FUNCTORS). Much of our discussion aboutuniversal properties can be cleanly expressed in terms of representable functors.(a) Suppose A and B are objects in a category C. Give a bijection between the nat-ural transformations hA ! hB of covariant functors C ! Sets (see Exercise 2.2.14for the definition) and the morphisms B ! A.(b) State the corresponding fact for contravariant functors hA (see Exercise 2.2.20).Remark: a contravariant functor F from C to sets is said to be representable if there is

a natural isomorphism % : F" $$ hA . This exercise shows that the representing

object A is determined up to unique isomorphism by the pair (F, %). There is a sim-ilar definition for covariant functors. (We will revisit this in §7.6, and this problemwill appear again as Exercise 7.6.B.)(c) Yoneda’s lemma. Suppose F is a covariant functor C ! Sets, and A " C. Givea bijection between the natural transformations hA ! F and F(A). State the corre-sponding fact for contravariant functors.

2.4 Limits and colimits

Limits and colimits provide two important examples defined by universalproperties. They generalize a number of familiar constructions. I’ll give the defi-nition first, and then show you why it is familiar. For example, fractions will bemotivating examples of colimits (Exercise 2.4.B(a)), and the p-adic numbers (Ex-ample 2.4.3) will be motivating examples of limits.

2.4.1. Limits. We say that a category is a small category if the objects and themorphisms are sets. (This is a technical condition intended only for experts.) Sup-pose I is any small category, and C is any category. Then a functor F : I ! C (i.e.with an object Ai " C for each element i " I, and appropriate commuting mor-phisms dictated by I) is said to be a diagram indexed by I. We call I an indexcategory. Our index categories will be partially ordered sets (Example 2.2.8), inwhich in particular there is at most one morphism between any two objects. (But

August 26, 2010. 29

other examples are sometimes useful.) For example, if ! is the category

•

%%

$$ •

%%• $$ •

and A is a category, then a functor ! ! A is precisely the data of a commutingsquare in A.

Then the limit is an object lim#(IAi of C along with morphisms fj : lim#(I

Ai !Aj such that if m : j ! k is a morphism in I, then

lim#(IAi

fj

%%

fk

''))))

))))

)

AjF(m) $$ Ak

commutes, and this object and maps to each Ai is universal (final) respect to thisproperty. More precisely, given any other object W along with maps gi : W ! Ai

commuting with the F(m) (if m : i ! j is a morphism in I, then gj = F(m) # gj),then there is a unique map g : W ! lim#(I

Ai so that gi = fi # g for all i. (In somecases, the limit is sometimes called the inverse limit or projective limit. We won’tuse this language.) By the usual universal property argument, if the limit exists, itis unique up to unique isomorphism.

2.4.2. Examples: products. For example, if I is the partially ordered set

•

%%• $$ •

we obtain the fibered product.If I is

• •we obtain the product.

If I is a set (i.e. the only morphisms are the identity maps), then the limit iscalled the product of the Ai, and is denoted

$i Ai. The special case where I has

two elements is the example of the previous paragraph.If I has an initial object e, then Ae is the limit, and in particular the limit

always exists.

2.4.3. Example: the p-adic numbers. The p-adic numbers, Zp, are often describedinformally (and somewhat unnaturally) as being of the form Zp = ? + ?p + ?p2 +?p3 + · · · . They are an example of a limit in the category of rings:

Zp

..****

****

**+++++++++++++++++

//,,,,,,,,,,,,,,,,,,,,,,,,,,,

· · · $$ Z/p3 $$ Z/p2 $$ Z/p

Limits do not always exist for any index category I. However, you can ofteneasily check that limits exist if the objects of your category can be interpreted as


sets with additional structure, and arbitrary products exist (respecting the set-likestructure).

2.4.A. IMPORTANT EXERCISE. Show that in the category Sets,%

(ai)i%I "&

i

Ai : F(m)(ai) = aj for all m " MorI(i, j) " Mor(I)

'

,

along with the obvious projection maps to each Ai, is the limit lim#(IAi.

This clearly also works in the category ModA of A-modules, and its specializa-tions such as Veck and Ab.

From this point of view, 2 + 3p + 2p2 + · · · " Zp can be understood as thesequence (2, 2 + 3p, 2 + 3p + 2p2, . . . ).

2.4.4. Colimits. More immediately relevant for us will be the dual (arrow-reversed version) of the notion of limit (or inverse limit). We just flip all the arrowsin that definition, and get the notion of a colimit. Again, if it exists, it is unique upto unique isomorphism. (In some cases, the colimit is sometimes called the directlimit, inductive limit, or injective limit. We won’t use this language. I prefer us-ing limit/colimit in analogy with kernel/cokernel and product/coproduct. Thisis more than analogy, as kernels and products may be interpreted as limits, andsimilarly with cokernels and coproducts. Also, I remember that kernels “map to”,and cokernels are “mapped to”, which reminds me that a limit maps to all the ob-jects in the big commutative diagram indexed by I; and a colimit has a map fromall the objects.)

Even though we have just flipped the arrows, colimits behave quite differentlyfrom limits.

2.4.5. Example. The ring 5!( Z of rational numbers whose denominators arepowers of 5 is a colimit lim(! 5!iZ. More precisely, 5( Z is the colimit of

Z $$ 5!1Z $$ 5!2Z $$ · · ·

The colimit over an index set I is called the coproduct, denoted"

i Ai, and isthe dual (arrow-reversed) notion to the product.

2.4.B. EXERCISE. (a) Interpret the statement “Q = lim(!1nZ”. (b) Interpret the

union of the some subsets of a given set as a colimit. (Dually, the intersection canbe interpreted as a limit.) The objects of the category in question are the subsets ofthe given set.

Colimits don’t always exist, but there are two useful large classes of examplesfor which they do.

2.4.6. Definition. A partially ordered set (S,%) is filtered (or is said to be a filteredset) if for each x, y " S, there is a z such that x % z and y % z. More generally, acategory I is filtered if:

(i) for each x, y " I, there is a z " I and arrows x ! z and y ! z, and(ii) for every two arrows u, v : x ! y, there is an arrow w : y ! z such that

w # u = w # v.

August 26, 2010. 31

2.4.C. EXERCISE. Suppose I is filtered. (We will be almost exclusively using thecase where I is a filtered set.) Show that any diagram in Sets indexed by I has thefollowing as a colimit:

%

a ")

i%I

Ai

'

/ (ai " Ai) ! f(ai) " Aj for every f : Ai ! Aj in the diagram.

This idea applies to many categories whose objects can be interpreted as setswith additional structure (such as abelian groups, A-modules, groups, etc.). Forexample, in Example 2.4.5, each element of the colimit is an element of somethingupstairs, but you can’t say in advance what it is an element of. For example, 17/125is an element of the 5!3Z (or 5!4Z, or later ones), but not 5!2Z. More generally,in the category of A-modules ModA, each element a of the colimit lim(! Ai can beinterpreted as an element of some a " Ai. The element a " lim(!Ai is 0 if there issome m : i ! j such that F(m)(a) = 0 (i.e. if it becomes 0 “later in the diagram”).Furthermore, two elements interpreted as ai " Ai and aj " Aj are the same ifthere are some arrows m : i ! k and n : j ! k such that F(m)(ai) = F(n)(aj), i.e.if they become the same “later in the diagram”. To add two elements interpretedas ai " Ai and aj " Aj, we choose arrows m : i ! k and n : j ! k, and theninterpret their sum as F(m)(ai) + F(n)(aj).

2.4.D. EXERCISE. Verify that the A-module described above is indeed the colimit.

2.4.E. USEFUL EXERCISE (LOCALIZATION AS COLIMIT). Generalize Exercise 2.4.B(a)to interpret localization of a ring as a colimit over a filtered set: suppose S is a mul-tiplicative set of A, and interpret S!1A = lim(!

1sA where the limit is over s " S.

A variant of this construction works without the filtered condition, if you haveanother means of “connecting elements in different objects of your diagram”. Forexample:

2.4.F. EXERCISE: COLIMITS OF A-MODULES WITHOUT THE FILTERED CONDITION.Suppose you are given a diagram of A-modules indexed by I: F : I ! ModA,where we let Ai := F(i). Show that the colimit is )i%IAi modulo the relationsaj ! F(m)(ai) for every m : i ! j in I (i.e. for every arrow in the diagram).

The following exercise shows that you have to be careful to remember whichcategory you are working in.

2.4.G. UNIMPORTANT EXERCISE. Consider the filtered set of abelian groupsp!nZp/Zp. Show that this system has colimit Qp/Zp in the category of abeliangroups, and the colimit 0 in the category of finite abelian groups. Here Qp is thefraction field of Zp, which can be interpreted as *p!nZp.

2.4.7. Summary. One useful thing to informally keep in mind is the following. Ina category where the objects are “set-like”, an element of a limit can be thought ofas an element in each object in the diagram, that are “compatible” (Exercise 2.4.A).And an element of a colimit can be thought of (“has a representative that is”) an ele-ment of a single object in the diagram (Exercise 2.4.C). Even though the definitions


of limit and colimit are the same, just with arrows reversed, these interpretationsare quite different.

2.5 Adjoints

We next come to an very useful construction closely related to universal prop-erties. Just as a universal property “essentially” (up to unique isomorphism) de-termines an object in a category (assuming such an object exists), “adjoints” es-sentially determine a functor (again, assuming it exists). Two covariant functorsF : A ! B and G : B ! A are adjoint if there is a natural bijection for all A " Aand B " B(2.5.0.1) &AB : MorB(F(A), B) ! MorA(A,G(B)).

We say that (F,G) form an adjoint pair, and that F is left-adjoint to G (and G isright-adjoint to F). By “natural” we mean the following. For all f : A ! A ! in A,we require

(2.5.0.2) MorB(F(A !), B)Ff"

$$

#A !B

%%

MorB(F(A), B)

#AB

%%MorA(A !, G(B))

f"$$ MorA(A,G(B))

to commute, and for all g : B ! B ! in B we want a similar commutative diagram tocommute. (Here f& is the map induced by f : A ! A !, and Ff& is the map inducedby Ff : L(A) ! L(A !).)

2.5.A. EXERCISE. Write down what this diagram should be. (Hint: do it byextending diagram (2.5.0.2) above.)

2.5.B. EXERCISE. Show that the map &AB (2.5.0.1) is given as follows. For each Athere is a map 'A : A ! GF(A) so that for any g : F(a) ! B, the correspondingf : A ! G(B) is given by the composition

A$A $$ GF(A)

Gg $$ G(B).

Similarly, there is a map (B : B ! FG(B) for each B so that for any f : A ! G(B),the corresponding map g : F(A) ! B is given by the composition

F(A)Ff $$ FG(B)

%B $$ B.

Here is an example of an adjoint pair.

2.5.C. EXERCISE. Suppose M, N, and P are A-modules. Describe a bijectionMorA(M'A N,P) * MorA(M, MorA(N,P)). (Hint: try to use the universal prop-erty.)

2.5.D. EXERCISE. Show that ·'A N and MorA(N, ·) are adjoint functors.

2.5.1. ! Fancier remarks we won’t use. You can check that the left adjoint deter-mines the right adjoint up to unique natural isomorphism, and vice versa, by a

August 26, 2010. 33

universal property argument. The maps 'A and (B of Exercise 2.5.B are calledthe unit and counit of the adjunction. This leads to a different characterization ofadjunction. Suppose functors F : A ! B and G : B ! A are given, along withnatural transformations ( : FG ! id and ' : id ! GF with the property thatG( # 'G = idG (for each B " G, the composition of 'G(B) : G(B) ! GFG(B) andG('B) : GFG(B) ! G(B) is the identity) and 'F # F( = idF. Then you can checkthat F is left adjoint to G. These facts aren’t hard to check, so if you want to usethem, you should verify everything for yourself.

2.5.2. Examples from other fields. For those familiar with representation theory:Frobenius reciprocity may be understood in terms of adjoints. Suppose V is afinite-dimensional representation of a finite group G, and W is a representation of

a subgroup H < G. Then induction and restriction are an adjoint pair (IndGH, ResG

H)between the category of G-modules and the category of H-modules.

Topologists’ favorite adjoint pair may be the suspension functor and the loopspace functor.

2.5.3. Example: groupification. Here is another motivating example: gettingan abelian group from an abelian semigroup. An abelian semigroup is just likean abelian group, except you don’t require an inverse. One example is the non-negative integers 0, 1, 2, . . . under addition. Another is the positive integers un-der multiplication 1, 2, . . . . From an abelian semigroup, you can create an abeliangroup. Here is a formalization of that notion. If S is a semigroup, then its groupi-fication is a map of semigroups $ : S ! G such that G is a group, and any othermap of semigroups from S to a group G ! factors uniquely through G.

S $$

"

!!---

----

- G

"!

%%G !

2.5.E. EXERCISE. Construct groupification H from the category of abelian semi-groups to the category of abelian groups. (One possibility of a construction: givenan abelian semigroup S, the elements of its groupification H(S) are (a, b), whichyou may think of as a ! b, with the equivalence that (a, b) ! (c, d) if a + d + e =b + c + e for some e " S. Describe addition in this group, and show that it satisfiesthe properties of an abelian group. Describe the semigroup map S ! H(S).) Let Fbe the forgetful morphism from the category of abelian groups Ab to the categoryof abelian semigroups. Show that H is left-adjoint to F.

(Here is the general idea for experts: We have a full subcategory of a category;this is called a “reflective subcategory”. We want to “project” from the category tothe subcategory. We have

Morcategory(S,H) = Morsubcategory(G,H)

automatically; thus we are describing the left adjoint to the forgetful functor. Howthe argument worked: we constructed something which was in the smaller cate-gory, which automatically satisfies the universal property.)

2.5.F. EXERCISE. Show that if a semigroup is already a group then groupificationis the identity morphism, by the universal property.


2.5.G. EXERCISE. The purpose of this exercise is to give you some practice with“adjoints of forgetful functors”, the means by which we get groups from semi-groups, and sheaves from presheaves. Suppose A is a ring, and S is a multiplica-tive subset. Then S!1A-modules are a fully faithful subcategory of the categoryof A-modules (meaning: the objects of the first category are a subset of the ob-jects of the second; and the morphisms between any two objects of the secondthat are secretly objects of the first are just the morphisms from the first). ThenM ! S!1M satisfies a universal property. Figure out what the universal propertyis, and check that it holds. In other words, describe the universal property enjoyedby M ! S!1M, and prove that it holds.

(Here is the larger story. Every S!1A-module is an A-module, and this is aninjective map, so we have a covariant forgetful functor F : ModS!1A ! ModA. Infact this is a fully faithful functor: it is injective on objects, and the morphismsbetween any two S!1A-modules as A-modules are just the same when they are con-sidered as S!1A-modules. Then there is a functor G : ModA ! ModS!1A, whichmight reasonably be called “localization with respect to S”, which is left-adjointto the forgetful functor. Translation: If M is an A-module, and N is an S!1A-module, then Mor(GM,N) (morphisms as S!1A-modules, which are the same asmorphisms as A-modules) are in natural bijection with Mor(M,FN) (morphismsas A-modules).)

Here is a table of adjoints that will come up for us.

situation category category left-adjoint right-adjointA B F : A ! B G : B ! A

A-modules ·'A N HomA(N, ·)ring maps A ! B ModA ModB ·AB forgetful

(extension of scalars) (restriction of scalars)(pre)sheaves on a presheaves sheaves on X sheafification forgetfultopological space X on X

(semi)groups semigroups groups groupification forgetfulsheaves, f : X ! Y sheaves on Y sheaves on X f!1 f&sheaves of abeliangroups or O-modules, sheaves on U sheaves on Y f! f!1

open immersionsf : U "! Y

quasicoherent quasicoherent quasicoherent f& f&sheaves, f : X ! Y sheaves on Y sheaves on X

2.5.4. Useful comment for experts. One last comment only for people who have seenadjoints before: If (F,G) is an adjoint pair of functors, then F commutes with col-imits, and G commutes with limits. Also, limits commute with limits and colimitscommute with colimits. We will prove these facts (and a little more) in §2.6.10.

2.6 Kernels, cokernels, and exact sequences: A brief introductionto abelian categories

August 26, 2010. 35

Since learning linear algebra, you have been familiar with the notions and be-haviors of kernels, cokernels, etc. Later in your life you saw them in the category ofabelian groups, and later still in the category of A-modules. Each of these notionsgeneralizes the previous one.

We will soon define some new categories (certain sheaves) that will have familiar-looking behavior, reminiscent of that of modules over a ring. The notions of ker-nels, cokernels, images, and more will make sense, and they will behave “the waywe expect” from our experience with modules. This can be made precise throughthe notion of an abelian category. Abelian categories are the right general settingin which one can do “homological algebra”, in which notions of kernel, cokernel,and so on are used, and one can work with complexes and exact sequences.

We will see enough to motivate the definitions that we will see in general:monomorphism (and subobject), epimorphism, kernel, cokernel, and image. Butin these notes we will avoid having to show that they behave “the way we expect”in a general abelian category because the examples we will see are directly inter-pretable in terms of modules over rings. In particular, it is not worth memorizingthe definition of abelian category.

Two central examples of an abelian category are the category Ab of abeliangroups, and the category ModA of A-modules. The first is a special case of thesecond (just take A = Z). As we give the definitions, you should verify that ModA

is an abelian category.We first define the notion of additive category. We will use it only as a stepping

stone to the notion of an abelian category.

2.6.1. Definition. A category C is said to be additive if it satisfies the followingproperties.

Ad1. For each A,B " C, Mor(A,B) is an abelian group, such that compositionof morphisms distributes over addition. (You should think about whatthis means — it translates to two distinct statements).

Ad2. C has a zero object, denoted 0. (This is an object that is simultaneously aninitial object and a final object, Defn. 2.3.3.)

Ad3. It has products of two objects (a product A $ B for any pair of objects),and hence by induction, products of any finite number of objects.

In an additive category, the morphisms are often called homomorphisms, andMor is denoted by Hom. In fact, this notation Hom is a good indication that you’reworking in an additive category. A functor between additive categories preservingthe additive structure of Hom, is called an additive functor.

2.6.2. Remarks. It is a consequence of the definition of additive category that finitedirect products are also finite direct sums (coproducts) — the details don’t matterto us. The symbol ) is used for this notion. Also, it is quick to show that additivefunctors send zero objects to zero objects (show that a is a 0-object if and only ifida = 0a; additive functors preserve both id and 0), and preserves products.

One motivation for the name 0-object is that the 0-morphism in the abeliangroup Hom(A,B) is the composition A ! 0 ! B.

Real (or complex) Banach spaces are an example of an additive category. Thecategory of free A-modules is another. The category ModA of A-modules is also anexample, but it has even more structure, which we now formalize as an exampleof an abelian category.


2.6.3. Definition. Let C be an additive category. A kernel of a morphismf : B ! C is a map i : A ! B such that f # i = 0, and that is universal with respectto this property. Diagramatically:

Z

++((

((

((

(

0

&&..............

"!

%%A

i $$

0

00Bf $$ C

(Note that the kernel is not just an object; it is a morphism of an object to B.) Henceit is unique up to unique isomorphism by universal property nonsense. A coker-nel is defined dually by reversing the arrows — do this yourself. The kernel off : B ! C is the limit (§2.4) of the diagram

0

%%B

f $$ C

and similarly the cokernel is a colimit.A morphism i : A ! B in C is monic if for all g : C ! A such that i # g = 0, we

have g = 0:C

!g=0

%%

0

++((

((

((

(

Ai $$ B

(Once we know what an abelian category is, you may check that a monic mor-phism in an abelian category is a monomorphism.) If i : A ! B is monic, thenwe say that A is a subobject of B, where the map i is implicit. Dually, there is thenotion of epi — reverse the arrows to find out what that is. The notion of quotientobject is defined dually to subobject.

An abelian category is an additive category satisfying three additional prop-erties.

(1) Every map has a kernel and cokernel.(2) Every monic morphism is the kernel of its cokernel.(3) Every epi morphism is the cokernel of its kernel.

It is a non-obvious (and imprecisely stated) fact that every property you wantto be true about kernels, cokernels, etc. follows from these three.

The image of a morphism f : A ! B is defined as im(f) = ker(coker f). It isthe unique factorization

Aepi

$$ im(f)monic $$ B

It is the cokernel of the kernel, and the kernel of the cokernel. The reader maywant to verify this as an exercise. It is unique up to unique isomorphism.

We will leave the foundations of abelian categories untouched. The key thingto remember is that if you understand kernels, cokernels, images and so on inthe category of modules over a ring ModA, you can manipulate objects in anyabelian category. This is made precise by Freyd-Mitchell Embedding Theorem.(The Freyd-Mitchell Embedding Theorem: If A is an abelian category such that

August 26, 2010. 37

Hom(a, a !) is a set for all a, a ! " A, then there is a ring A and an exact, full faith-ful functor from A into ModA, which embeds A as a full subcategory. A proofis sketched in [W, §1.6], and references to a complete proof are given there. Themoral is that to prove something about a diagram in some abelian category, wemay pretend that it is a diagram of modules over some ring, and we may then“diagram-chase” elements. Moreover, any fact about kernels, cokernels, and soon that holds in ModA holds in any abelian category.) However, the abelian cate-gories we’ll come across will obviously be related to modules, and our intuitionwill clearly carry over, so we needn’t invoke a theorem whose proof we haven’tread. For example, we’ll show that sheaves of abelian groups on a topologicalspace X form an abelian category (§3.5), and the interpretation in terms of “com-patible germs” will connect notions of kernels, cokernels etc. of sheaves of abeliangroups to the corresponding notions of abelian groups.

2.6.4. Complexes, exactness, and homology.We say a sequence

(2.6.4.1) Af $$ B

g $$ C

is a complex if g # f = 0, and is exact if ker g = im f. An exact sequence withfive terms, the first and last of which are 0, is a short exact sequence. Note that

Af $$ B $$ C $$ 0 being exact is equivalent to describing C as a cokernel

of f (with a similar statement for 0 $$ A $$ Bg $$ C ).

If (2.6.4.1) is a complex, then its homology (often denoted H) is ker g / im f. Wesay that the ker g are the cycles, and im f are the boundaries (so homology is “cy-cles mod boundaries”). If the complex is indexed in decreasing order, the indicesare often written as subscripts, and Hi is the homology at Ai+1 ! Ai ! Ai!1. Ifthe complex is indexed in increasing order, the indices are often written as super-scripts, and the homology Hi at Ai!1 ! Ai ! Ai+1 is often called cohomology.

An exact sequence

(2.6.4.2) A• : · · · $$ Ai!1fi!1

$$ Aifi

$$ Ai+1fi+1

$$ · · ·

can be “factored” into short exact sequences

0 $$ ker fi $$ Ai $$ ker fi+1 $$ 0

which is helpful in proving facts about long exact sequences by reducing them tofacts about short exact sequences.

More generally, if (2.6.4.2) is assumed only to be a complex, then it can be“factored” into short exact sequences.

(2.6.4.3) 0 $$ ker fi $$ Ai $$ im fi $$ 0

0 $$ im fi!1 $$ ker fi $$ Hi(A•) $$ 0


2.6.A. EXERCISE. Describe exact sequences

(2.6.4.4) 0 $$ im fi $$ Ai+1 $$ coker fi $$ 0

0 $$ Hi(A•) $$ coker fi!1 $$ im fi $$ 0

(These are somehow dual to (2.6.4.3). In fact in some mirror universe this mighthave been given as the standard definition of homology.)

2.6.B. EXERCISE. Suppose

0d0

$$ A1d1

$$ · · · dn!1$$ An dn

$$$$ 0

is a complex of finite-dimensional k-vector spaces (often called A• for short). Showthat

+(!1)i dim Ai =

+(!1)ihi(A•). (Recall that hi(A•) = dim ker(di)/ im(di!1).)

In particular, if A• is exact, then+

(!1)i dim Ai = 0. (If you haven’t dealt muchwith cohomology, this will give you some practice.)

2.6.C. IMPORTANT EXERCISE. Suppose C is an abelian category. Define the cate-gory ComC as follows. The objects are infinite complexes

A• : · · · $$ Ai!1fi!1

$$ Aifi

$$ Ai+1fi+1

$$ · · ·

in C, and the morphisms A• ! B• are commuting diagrams

(2.6.4.5) A• :

%%

· · · $$ Ai!1

%%

fi!1$$ Ai

fi$$

%%

Ai+1fi+1

$$

%%

· · ·

B• : · · · $$ Bi!1gi!1

$$ Bigi

$$ Bi+1gi+1

$$ · · ·

Show that ComC is an abelian category. (Feel free to deal with the special caseModA.)

2.6.D. IMPORTANT EXERCISE. Show that (2.6.4.5) induces a map of homologyH(Ai) ! H(Bi). (Again, feel free to deal with the special case ModA.)

We will later define when two maps of complexes are homotopic (§23.1), andshow that homotopic maps induce isomorphisms on cohomology (Exercise 23.1.A),but we won’t need that any time soon.

August 26, 2010. 39

2.6.5. Theorem (Long exact sequence). — A short exact sequence of complexes

0• :

%%

· · · $$ 0 $$

%%

0 $$

%%

0 $$

%%

· · ·

A• :

%%

· · · $$ Ai!1

%%

fi!1$$ Ai

fi$$

%%

Ai+1fi+1

$$

%%

· · ·

B• :

%%

· · · $$ Bi!1

%%

gi!1

$$ Bigi

$$

%%

Bi+1gi+1

$$

%%

· · ·

C• :

%%

· · · $$ Ci!1hi!1

$$

%%

Cihi

$$

%%

Ci+1hi+1

$$

%%

· · ·

0• : · · · $$ 0 $$ 0 $$ 0 $$ · · ·

induces a long exact sequence in cohomology

. . . $$ Hi!1(C•) $$

Hi(A•) $$ Hi(B•) $$ Hi(C•) $$

Hi+1(A•) $$ · · ·

(This requires a definition of the connecting homomorphism Hi!1(C•) ! Hi(A•),which is natural in an appropriate sense.) For a concise proof in the case of com-plexes of modules, and a discussion of how to show this in general, see [W, §1.3]. Itwill also come out of our discussion of spectral sequences as well (again, in the cat-egory of modules over a ring), see Exercise 2.7.E, but this is a somewhat perverseway of proving it.

2.6.6. Exactness of functors. If F : A ! B is a covariant additive functor from oneabelian category to another, we say that F is right-exact if the exactness of

A ! $$ A $$ A !! $$ 0,

in A implies that

F(A !) $$ F(A) $$ F(A !!) $$ 0

is also exact. Dually, we say that F is left-exact if the exactness of

0 $$ A ! $$ A $$ A !! implies

0 $$ F(A !) $$ F(A) $$ F(A !!) is exact.


A contravariant functor is left-exact if the exactness of

A ! $$ A $$ A !! $$ 0 implies

0 $$ F(A !!) $$ F(A) $$ F(A !) is exact.

The reader should be able to deduce what it means for a contravariant functor tobe right-exact.

A covariant or contravariant functor is exact if it is both left-exact and right-exact.

2.6.E. EXERCISE. Suppose F is an exact functor. Show that applying F to an exactsequence preserves exactness. For example, if F is covariant, and A ! ! A ! A !!

is exact, then FA ! ! FA ! FA !! is exact. (This will be generalized in Exer-cise 2.6.H(c).)

2.6.F. EXERCISE. Suppose A is a ring, S + A is a multiplicative subset, and M isan A-module.(a) Show that localization of A-modules ModA ! ModS!1A is an exact covariantfunctor.(b) Show that · ' M is a right-exact covariant functor ModA ! ModA. (This is arepeat of Exercise 2.3.G.)(c) Show that Hom(M, ·) is a left-exact covariant functor ModA ! ModA.(d) Show that Hom(·,M) is a left-exact contravariant functor ModA ! ModA.

2.6.G. EXERCISE. Suppose M is a finitely presented A-module: M has a finitenumber of generators, and with these generators it has a finite number of relations;or usefully equivalently, fits in an exact sequence

(2.6.6.1) A'q ! A'p ! M ! 0

Use (2.6.6.1) and the left-exactness of Hom to describe an isomorphism

S!1 HomA(M,N) != HomS!1A(S!1M,S!1N).

(You might be able to interpret this in light of a variant of Exercise 2.6.H below, forleft-exact contravariant functors rather than right-exact covariant functors.)

2.6.7. ! Two useful facts in homological algebra.We now come to two (sets of) facts I wish I had learned as a child, as they

would have saved me lots of grief. They encapsulate what is best and worst ofabstract nonsense. The statements are so general as to be nonintuitive. The proofsare very short. They generalize some specific behavior it is easy to prove in anad hoc basis. Once they are second nature to you, many subtle facts will be comeobvious to you as special cases. And you will see that they will get used (implicitlyor explicitly) repeatedly.

2.6.8. ! Interaction of homology and (right/left-)exact functors.You might wait to prove this until you learn about cohomology in Chapter 20,

when it will first be used in a serious way.

August 26, 2010. 41

2.6.H. IMPORTANT EXERCISE (THE FHHF THEOREM). This result can take you far,and perhaps for that reason it has sometimes been called the fernbahnhof (Fern-baHnHoF) theorem, [N, Ex. 2.6.H]. Suppose F : A ! B is a covariant functor ofabelian categories. Suppose C• is a complex in A.

(a) (F right-exact yields FH• $$ H• ) If F is right-exact, describe a naturalmorphism FH• ! H•F. (More precisely, for each i, the left side is F ap-plied to the cohomology at piece i of C•, while the right side is the coho-mology at piece i of FC•.)

(b) (F left-exact yields FH• H•,, ) If F is right-exact, describe a naturalmorphism FH• ! H•F. (More precisely, for each i, the left side is F ap-plied to the cohomology at piece i of C•, while the right side is the coho-mology at piece i of FC•.)

(c) (F exact yields FH• ,, $$ H• ) If F is exact, show that the morphisms of (a)and (b) are inverses and thus isomorphisms.

Hint for (a): use Cp dp$$ Cp+1 $$ coker dp $$ 0 to give an isomorphism

F coker dp != coker Fdp. Then use the first line of (2.6.4.4) to give a surjection

F im dp $$ $$ im Fdp . Then use the second line of (2.6.4.4) to give the desired

map FHpC• $$ HpFC• . While you are at it, you may as well describe a map

for the fourth member of the quartet {ker, coker, im, H, }: F ker dp $$ ker Fdp .

2.6.9. If this makes your head spin, you may prefer to think of it in the followingspecific case, where both A and B are the category of A-modules, and F is ·'N forsome fixed N-module. Your argument in this case will translate without changeto yield a solution to Exercise 2.6.H(a) and (c) in general. If 'N is exact, then N iscalled a flat A-module. (The notion of flatness will turn out to be very important,and is discussed in detail in Chapter 24.)

For example, localization is exact, so S!1A is a flat A-algebra for all multiplica-tive sets S. Thus taking cohomology of a complex of A-modules commutes withlocalization — something you could verify directly.

2.6.10. ! Interaction of adjoints, (co)limits, and (left- and right-) exactness.A surprising number of arguments boil down the statement:Limits (e.g. kernels) commute with limits and right-adjoints. In particular, both right-

adjoints and limits are left exact.as well as its dual:Colimits (e.g. cokernels) commute with colimits and left-adjoints. In particular, both

left-adjoints and colimits are left exact.These statements were promised in §2.5.4. The latter has a useful extension:In an abelian category, colimits over filtered index categories are exact.(“Filtered” was defined in §2.4.6.) If you want to use these statements (for

example, later in these notes), you will have to prove them. Let’s now make themprecise.

2.6.I. EXERCISE (KERNELS COMMUTE WITH LIMITS). Suppose C is an abeliancategory, and a : I ! C and b : I ! C are two diagrams in C indexed by I. Forconvenience, let Ai = a(i) and Bi = b(i) be the objects in those two diagrams. Let


hi : Ai ! Bi be maps commuting with the maps in the diagram. (Translation: his a natural transformation of functors a ! b, see §2.2.21.) Then the ker hi formanother diagram in I indexed by I. Describe a natural isomorphism lim#(ker hi

!=ker(lim#(Ai ! lim#(Bi).

2.6.J. EXERCISE. Make sense of the statement that “limits commute with limits” ina general category, and prove it. (Hint: recall that kernels are limits. The previousexercise should be a corollary of this one.)

2.6.11. Proposition (right-adjoints commute with limits). — Suppose (F : C !D, G : D ! C) is a pair of adjoint functors. If A = lim#(Ai is a limit in D of a diagramindexed by I, then GA = lim#(GAi (with the corresponding maps GA ! GAi) is a limitin C.

Proof. We must show that GA ! GAi satisfies the universal property of limits.Suppose we have maps W ! GAi commuting with the maps of I. We wish toshow that there exists a unique W ! GA extending the W ! GAi. By adjointnessof F and G, we can restate this as: Suppose we have maps FW ! Ai commutingwith the maps of I. We wish to show that there exists a unique FW ! A extendingthe FW ! Ai. But this is precisely the universal property of the limit. !

Of course, the dual statements to Exercise 2.6.J and Proposition 2.6.11 hold bythe dual arguments.

If F and G are additive functors between abelian categories, then (as kernelsare limits and cokernels are colimits) G is left-exact and F is right-exact.

2.6.K. EXERCISE. Show that in an abelian category, colimits over filtered indexcategories are exact. Right-exactness follows from the above discussion, so theissue is left-exactness. (Possible hint: After you show that localization is exact,Exercise 2.6.F(a), or sheafification is exact, Exercise 3.5.D, in a hands on way, youwill be easily able to prove this. Conversely, this exercise will quickly imply thosetwo.)

2.6.L. EXERCISE. Show that filtered colimits commute with homology. Hint: usethe FHHF Theorem (Exercise 2.6.H), and the previous Exercise.

2.6.12. ! Dreaming of derived functors. When you see a left-exact functor, youshould always dream that you are seeing the end of a long exact sequence. If

0 ! M ! ! M ! M !! ! 0

is an exact sequence in abelian category A, and F : A ! B is a left-exact functor,then

0 ! FM ! ! FM ! FM !!

is exact, and you should always dream that it should continue in some naturalway. For example, the next term should depend only on M !, call it R1FM !, and if itis zero, then FM ! FM !! is an epimorphism. This remark holds true for left-exactand contravariant functors too. In good cases, such a continuation exists, and isincredibly useful. We will discuss this in Chapter 23.

August 26, 2010. 43

2.7 ! Spectral sequences

Spectral sequences are a powerful book-keeping tool for proving things in-volving complicated commutative diagrams. They were introduced by Leray inthe 1940’s at the same time as he introduced sheaves. They have a reputation forbeing abstruse and difficult. It has been suggested that the name ‘spectral’ wasgiven because, like spectres, spectral sequences are terrifying, evil, and danger-ous. I have heard no one disagree with this interpretation, which is perhaps notsurprising since I just made it up.

Nonetheless, the goal of this section is to tell you enough that you can usespectral sequences without hesitation or fear, and why you shouldn’t be frightenedwhen they come up in a seminar. What is perhaps different in this presentationis that we will use spectral sequence to prove things that you may have alreadyseen, and that you can prove easily in other ways. This will allow you to getsome hands-on experience for how to use them. We will also see them only in thespecial case of double complexes (which is the version by far the most often usedin algebraic geometry), and not in the general form usually presented (filteredcomplexes, exact couples, etc.). See [W, Ch. 5] for more detailed information ifyou wish.

You should not read this section when you are reading the rest of Chapter 2.Instead, you should read it just before you need it for the first time. When youfinally do read this section, you must do the exercises.

For concreteness, we work in the category Veck of vector spaces over a fieldk. However, everything we say will apply in any abelian category, such as thecategory ModA of A-modules.

2.7.1. Double complexes.A double complex is a collection of vector spaces Ep,q (p, q " Z), and “right-

ward” morphisms dp,q! : Ep,q ! Ep,q+1 and “upward” morphisms dp,q

, : Ep,q !Ep+1,q. In the superscript, the first entry denotes the row number, and the secondentry denotes the column number, in keeping with the convention for matrices,but opposite to how the (x, y)-plane is labeled. The subscript is meant to suggestthe direction of the arrows. We will always write these as d! and d, and ignorethe superscripts. We require that d! and d, satisfying (a) d2

! = 0, (b) d2, = 0,

and one more condition: (c) either d!d, = d,d! (all the squares commute) ord!d, + d,d! = 0 (they all anticommute). Both come up in nature, and you canswitch from one to the other by replacing dp,q

, with (!1)qdp,q, . So I will assume

that all the squares anticommute, but that you know how to turn the commutingcase into this one. (You will see that there is no difference in the recipe, basicallybecause the image and kernel of a homomorphism f equal the image and kernel


respectively of !f.)

Ep+1,qdp+1,q

! $$ Ep+1,q+1

anticommutes

Ep,q

dp,q

,

--

dp,q! $$ Ep,q+1

dp,q+1

,

--

There are variations on this definition, where for example the vertical arrowsgo downwards, or some different subset of the Ep,q are required to be zero, but Iwill leave these straightforward variations to you.

From the double complex we construct a corresponding (single) complex E•

with Ek = )iEi,k!i, with d = d! + d, . In other words, when there is a single

superscript k, we mean a sum of the kth antidiagonal of the double complex. Thesingle complex is sometimes called the total complex. Note that d2 = (d! +d,)

2 =d2

! + (d!d, + d,d! ) + d2, = 0, so E• is indeed a complex.

The cohomology of the single complex is sometimes called the hypercoho-mology of the double complex. We will instead use the phrase “cohomology ofthe double complex”.

Our initial goal will be to find the cohomology of the double complex. Youwill see later that we secretly also have other goals.

A spectral sequence is a recipe for computing some information about thecohomology of the double complex. I won’t yet give the full recipe. Surprisingly,this fragmentary bit of information is sufficent to prove lots of things.

2.7.2. Approximate Definition. A spectral sequence with rightward orientationis a sequence of tables or pages !Ep,q

0 , !Ep,q1 , !Ep,q

2 , . . . (p, q " Z), where !Ep,q0 =

Ep,q, along with a differential

!dp,qr : !Ep,q

r ! !Ep+r,q!r+1

with !dp,qr # !dp,q

r = 0, and with an isomorphism of the cohomology of !dr at

!Ep,q (i.e. ker !dp,qr / im !dp!r,q+r!1

r ) with !Ep,qr+1.

The orientation indicates that our 0th differential is the rightward one: d0 =d! . The left subscript “!” is usually omitted.

The order of the morphisms is best understood visually:

(2.7.2.1) •

•

•

•d0

$$

d1

--d2

11//////////////

d3

22&&&&&&&&&&&&&&&&&&&&&&&

•

August 26, 2010. 45

(the morphisms each apply to different pages). Notice that the map always is“degree 1” in the grading of the single complex E•.

The actual definition describes what E•,•r and d•,•

r really are, in terms of E•,•.We will describe d0, d1, and d2 below, and you should for now take on faith thatthis sequence continues in some natural way.

Note that Ep,qr is always a subquotient of the corresponding term on the 0th

page Ep,q0 = Ep,q. In particular, if Ep,q = 0, then Ep,q

r = 0 for all r, so Ep,qr = 0

unless p, q " Z#0.Suppose now that E•,• is a first quadrant double complex, i.e. Ep,q = 0 for p <

0 or q < 0. Then for any fixed p, q, once r is sufficiently large, Ep,qr+1 is computed

from (E•,•r , dr) using the complex

0

Ep,qr

dp,qr

330000000000000

0

dp+r,q!r!1r

330000000000000

and thus we have canonical isomorphisms

Ep,qr

!= Ep,qr+1

!= Ep,qr+2

!= · · ·

We denote this module Ep,q( . The same idea works in other circumstances, for

example if the double complex is only nonzero in a finite number of rows — Ep,q =0 unless p0 < p < pq. This will come up for example in the long exact sequenceand mapping cone discussion (Exercises 2.7.E and 2.7.F below).

We now describe the first few pages of the spectral sequence explicitly. Asstated above, the differential d0 on E•,•

0 = E•,• is defined to be d! . The rows arecomplexes:

• $$ • $$ •

The 0th page E0: • $$ • $$ •

• $$ • $$ •

and so E1 is just the table of cohomologies of the rows. You should check that

there are now vertical maps dp,q1 : Ep,q

1 ! Ep+1,q1 of the row cohomology groups,

induced by d, , and that these make the columns into complexes. (This is essen-tially the fact that a map of complexes induces a map on homology.) We have


“used up the horizontal morphisms”, but “the vertical differentials live on”.

• • •

The 1st page E1: •

--

•

--

•

--

•

--

•

--

•

--

We take cohomology of d1 on E1, giving us a new table, Ep,q2 . It turns out that

there are natural morphisms from each entry to the entry two above and one to theleft, and that the composition of these two is 0. (It is a very worthwhile exerciseto work out how this natural morphism d2 should be defined. Your argumentmay be reminiscent of the connecting homomorphism in the Snake Lemma 2.7.5or in the long exact sequence in cohomology arising from a short exact sequenceof complexes, Exercise 2.6.C. This is no coincidence.)

• • •

The 2nd page E2: • • •

• •

11111111111111111

•

11111111111111111

This is the beginning of a pattern.Then it is a theorem that there is a filtration of Hk(E•) by Ep,q

( where p+q = k.(We can’t yet state it as an official Theorem because we haven’t precisely definedthe pages and differentials in the spectral sequence.) More precisely, there is afiltration

(2.7.2.2) E0,k(

! "E1,k!1( $$ ?

! "E2,k!2( $$ · · · ! " E0,k

$$ Hk(E•)

where the quotients are displayed above each inclusion. (I always forget whichway the quotients are supposed to go, i.e. whether Ek,0 or E0,k is the subobject.One way of remembering it is by having some idea of how the result is proved.)

We say that the spectral sequence !E•,•• converges to H•(E•). We often say

that !E•,•2 (or any other page) abuts to H•(E•).

Although the filtration gives only partial information about H•(E•), some-times one can find H•(E•) precisely. One example is if all Ei,k!i

( are zero, or ifall but one of them are zero (e.g. if Ei,k!i

r has precisely one non-zero row or col-umn, in which case one says that the spectral sequence collapses at the rth step,although we will not use this term). Another example is in the category of vectorspaces over a field, in which case we can find the dimension of Hk(E•). Also, inlucky circumstances, E2 (or some other small page) already equals E( .

2.7.A. EXERCISE: INFORMATION FROM THE SECOND PAGE. Show that H0(E•) =E0,0

( = E0,02 and

0 $$ E0,12

$$ H1(E•) $$ E1,02

d1,02 $$ E0,2

2$$ H2(E•).

August 26, 2010. 47

2.7.3. The other orientation.You may have observed that we could as well have done everything in the

opposite direction, i.e. reversing the roles of horizontal and vertical morphisms.Then the sequences of arrows giving the spectral sequence would look like this(compare to (2.7.2.1)).

(2.7.3.1) •

•

--

$$

&&..............

4422222222222222222222222 •

•

•This spectral sequence is denoted ,E

•,•• (“with the upwards orientation”). Then

we would again get pieces of a filtration of H•(E•) (where we have to be a bitcareful with the order with which ,E

p,q( corresponds to the subquotients — it in

the opposite order to that of (2.7.2.2) for !Ep,q( ). Warning: in general there is no

isomorphism between !Ep,q( and ,E

p,q( .

In fact, this observation that we can start with either the horizontal or verticalmaps was our secret goal all along. Both algorithms compute information aboutthe same thing (H•(E•)), and usually we don’t care about the final answer — weoften care about the answer we get in one way, and we get at it by doing thespectral sequence in the other way.

2.7.4. Examples.We are now ready to see how this is useful. The moral of these examples is

the following. In the past, you may have proved various facts involving varioussorts of diagrams, by chasing elements around. Now, you will just plug them intoa spectral sequence, and let the spectral sequence machinery do your chasing foryou.

2.7.5. Example: Proving the Snake Lemma. Consider the diagram

0 $$ D $$ E $$ F $$ 0

0 $$ A $$

&

--

B $$

'

--

C

(

--

$$ 0

where the rows are exact in the middle (at B, C, D, G, H, I) and the squares com-mute. (Normally the Snake Lemma is described with the vertical arrows pointingdownwards, but I want to fit this into my spectral sequence conventions.) We wishto show that there is an exact sequence

(2.7.5.1) 0 ! ker)! ker*! ker+! coker)! coker*! coker+! 0.

We plug this into our spectral sequence machinery. We first compute the co-homology using the rightwards orientation, i.e. using the order (2.7.2.1). Then be-cause the rows are exact, Ep,q

1 = 0, so the spectral sequence has already converged:Ep,q

( = 0.


We next compute this “0” in another way, by computing the spectral sequenceusing the upwards orientation. Then ,E

•,•1 (with its differentials) is:

0 $$ coker) $$ coker* $$ coker+ $$ 0

0 $$ ker) $$ ker* $$ ker+ $$ 0.

Then ,E•,•2 is of the form:

0

&&33333333333333 0

&&..............

0

&&.............. ??

&&33333333333333 ?

&&.............. ? 0

0 ? ?

&&.............. ??

&&33333333333333 0

0 0

We see that after ,E2, all the terms will stabilize except for the double-question-marks — all maps to and from the single question marks are to and from 0-entries.And after ,E3, even these two double-quesion-mark terms will stabilize. But inthe end our complex must be the 0 complex. This means that in ,E2, all the entriesmust be zero, except for the two double-question-marks, and these two must beisomorphic. This means that 0 ! ker)! ker*! ker+ and coker)! coker*!coker+ ! 0 are both exact (that comes from the vanishing of the single-question-marks), and

coker(ker*! ker+) != ker(coker)! coker*)

is an isomorphism (that comes from the equality of the double-question-marks).Taken together, we have proved the exactness of (2.7.5.1), and hence the SnakeLemma! (Notice: in the end we didn’t really care about the double complex. Wejust used it as a prop to prove the snake lemma.)

Spectral sequences make it easy to see how to generalize results further. Forexample, if A ! B is no longer assumed to be injective, how would the conclusionchange?

2.7.B. UNIMPORTANT EXERCISE (GRAFTING EXACT SEQUENCES, A WEAKER VER-SION OF THE SNAKE LEMMA). Extend the snake lemma as follows. Suppose wehave a commuting diagram

0 $$ X ! $$ Y ! $$ Z ! $$ A ! $$ · · ·

· · · $$ W $$

--

X $$

a

--

Y $$

b

--

Z $$

c

--

0.

--

August 26, 2010. 49

where the top and bottom rows are exact. Show that the top and bottom rows canbe ”grafted together” to an exact sequence

· · · $$ W $$ ker a $$ ker b $$ ker c

$$ coker a $$ coker b $$ coker c $$ A ! $$ · · · .

2.7.6. Example: the Five Lemma. Suppose

(2.7.6.1) F $$ G $$ H $$ I $$ J

A $$

&

--

B $$

'

--

C

(

--

$$ D $$

)

--

E

%

--

where the rows are exact and the squares commute.Suppose ), *, ,, ( are isomorphisms. We will show that + is an isomorphism.We first compute the cohomology of the total complex using the rightwards

orientation (2.7.2.1). We choose this because we see that we will get lots of zeros.Then !E•,•

1 looks like this:

? 0 0 0 ?

?

--

0

--

0

--

0

--

?

--

Then !E2 looks similar, and the sequence will converge by E2, as we will never getany arrows between two non-zero entries in a table thereafter. We can’t concludethat the cohomology of the total complex vanishes, but we can note that it van-ishes in all but four degrees — and most important, it vanishes in the two degreescorresponding to the entries C and H (the source and target of +).

We next compute this using the upwards orientation (2.7.3.1). Then ,E1 lookslike this:

0 $$ 0 $$ ? $$ 0 $$ 0

0 $$ 0 $$ ? $$ 0 $$ 0

and the spectral sequence converges at this step. We wish to show that those twoquestion marks are zero. But they are precisely the cohomology groups of the totalcomplex that we just showed were zero — so we’re done!

The best way to become comfortable with this sort of argument is to try itout yourself several times, and realize that it really is easy. So you should do thefollowing exercises!

2.7.C. EXERCISE: THE SUBTLE FIVE LEMMA. By looking at the spectral sequenceproof of the Five Lemma above, prove a subtler version of the Five Lemma, whereone of the isomorphisms can instead just be required to be an injection, and an-other can instead just be required to be a surjection. (I am deliberately not tellingyou which ones, so you can see how the spectral sequence is telling you how toimprove the result.)


2.7.D. EXERCISE. If * and , (in (2.7.6.1)) are injective, and ) is surjective, showthat + is injective. Give the dual statement (whose proof is of course essentiallythe same).

2.7.E. EXERCISE. Use spectral sequences to show that a short exact sequence ofcomplexes gives a long exact sequence in cohomology (Exercise 2.6.C).

2.7.F. EXERCISE (THE MAPPING CONE). Suppose µ : A• ! B• is a morphism ofcomplexes. Suppose C• is the single complex associated to the double complexA• ! B•. (C• is called the mapping cone of µ.) Show that there is a long exactsequence of complexes:

· · · ! Hi!1(C•) ! Hi(A•) ! Hi(B•) ! Hi(C•) ! Hi+1(A•) ! · · · .

(There is a slight notational ambiguity here; depending on how you index yourdouble complex, your long exact sequence might look slightly different.) In partic-ular, we will use the fact that µ induces an isomorphism on cohomology if and onlyif the mapping cone is exact. (We won’t use it until the proof of Theorem 20.2.4.)

The Grothendieck (or composition of functor) spectral sequence (Exercise 23.3.D)will be an important example of a spectral sequence that specializes in a numberof useful ways.

You are now ready to go out into the world and use spectral sequences to yourheart’s content!

2.7.7. !! Complete definition of the spectral sequence, and proof.You should most definitely not read this section any time soon after reading

the introduction to spectral sequences above. Instead, flip quickly through it toconvince yourself that nothing fancy is involved.

We consider the rightwards orientation. The upwards orientation is of coursea trivial variation of this.

2.7.8. Goals. We wish to describe the pages and differentials of the spectral se-quence explicitly, and prove that they behave the way we said they did. Moreprecisely, we wish to:

(a) describe Ep,qr ,

(b) verify that Hk(E•) is filtered by Ep,k!p( as in (2.7.2.2),

(c) describe dr and verify that d2r = 0, and

(d) verify that Ep,qr+1 is given by cohomology using dr.

Before tacking these goals, you can impress your friends by giving this shortdescription of the pages and differentials of the spectral sequence. We say thatan element of E•,• is a (p, q)-strip if it is an element of )l#0Ep+l,q!l (see Fig. 2.1).Its non-zero entries lie on a semi-infinite antidiagonal starting with position (p, q).We say that the (p, q)-entry (the projection to Ep,q) is the leading term of the (p, q)-

strip. Let Sp,q + E•,• be the submodule of all the (p, q)-strips. Clearly Sp,q +Ep+q, and S0,k = Ek.

Note that the differential d = d, +d! sends a (p, q)-strip x to a (p, q+ 1)-stripdx. If dx is furthermore a (p + r, q + r + 1)-strip (r " Z#0), we say that x is an

r-closed (p, q)-strip. We denote the set of such Sp,qr (so for example Sp,q

0 = Sp,q,

August 26, 2010. 51

. . . 0 0 0 0

0 ,p+2,q!2 0 0 0

0 0 ,p+1,q!1 0 0

0 0 0 ,p,q 0

0 0 0 0 0p!1,q+1

FIGURE 2.1. A (p, q)-strip (in Sp,q + Ep+q). Clearly S0,k = Ek.

and S0,k0 = Ek). An element of Sp,q

r may be depicted as:

. . .$$ ?

,p+2,q!2

--

$$ 0

,p+1,q!1

--

$$ 0

,p,q $$

--

0

2.7.9. Preliminary definition of Ep,qr . We are now ready to give a first definition of

Ep,qr , which by construction should be a subquotient of Ep,q = Ep,q

0 . We describeit as such by describing two submodules Yp,q

r + Xp,qr + Ep,q, and defining Ep,q

r =Xp,q

r /Yp,qr . Let Xp,q

r be those elements of Ep,q that are the leading terms of r-closed(p, q)-strips. Note that by definition, d sends (r ! 1)-closed Sp!(r!1),q+(r!1)!1-strips to (p, q)-strips. Let Yp,q

r be the leading ((p, q))-terms of the differential d of(r!1)-closed (p!(r!1), q+(r!1)!1)-strips (where the differential is consideredas a (p, q)-strip).

We next give the definition of the differential dr of such an element x " Xp,qr .

We take any r-closed (p, q)-strip with leading term x. Its differential d is a (p +r, q!r+1)-strip, and we take its leading term. The choice of the r-closed (p, q)-stripmeans that this is not a well-defined element of Ep,q. But it is well-defined modulothe (r ! 1)-closed (p + 1, r + 1)-strips, and hence gives a map Ep,q

r ! Ep+r,q!r+1r .

This definition is fairly short, but not much fun to work with, so we will forgetit, and instead dive into a snakes’ nest of subscripts and superscripts.


We begin with making some quick but important observations about (p, q)-strips.

2.7.G. EXERCISE. Verify the following.

(a) Sp,q = Sp+1,q!1 ) Ep,q.(b) (Any closed (p, q)-strip is r-closed for all r.) Any element x of Sp,q = Sp,q

0

that is a cycle (i.e. dx = 0) is automatically in Sp,qr for all r. For example,

this holds when x is a boundary (i.e. of the form dy).(c) Show that for fixed p, q,

Sp,q0 - Sp,q

1 - · · · - Sp,qr - · · ·

stabilizes for r . 0 (i.e. Sp,qr = Sp,q

r+1 = · · · ). Denote the stabilized mod-ule Sp,q

( . Show Sp,q( is the set of closed (p, q)-strips (those (p, q)-strips

annihilated by d, i.e. the cycles). In particular, S0,kr is the set of cycles in

Ek.

2.7.10. Defining Ep,qr .

Define Xp,qr := Sp,q

r /Sp+1,q!1r!1 and Y := dS

p!(r!1),q+(r!1)!1r!1 /Sp+1,q!1

r!1 .Then Yp,q

r + Xp,qr by Exercise 2.7.G(b). We define

(2.7.10.1) Ep,qr =

Xp,qr

Yp,qr

=Sp,q

r

dSp!(r!1),q+(r!1)!1r!1 + Sp+1,q!1

r!1

We have completed Goal 2.7.8(a).You are welcome to verify that these definitions of Xp,q

r and Yp,qr and hence

Ep,qr agree with the earlier ones of §2.7.9 (and in particular Xp,q

r and Yp,qr are both

submodules of Ep,q), but we won’t need this fact.

2.7.H. EXERCISE: Ep,k!p( GIVES SUBQUOTIENTS OF Hk(E•). By Exercise 2.7.G(c),

Ep,qr stabilizes as r ! (. For r . 0, interpret Sp,q

r /dSp!(r!1),q+(r!1)!1r!1 as the

cycles in Sp,q( + Ep+q modulo those boundary elements of dEp+q!1 contained in

Sp,q( . Finally, show that Hk(E•) is indeed filtered as described in (2.7.2.2).

We have completed Goal 2.7.8(b).

2.7.11. Definition of dr.We shall see that the map dr : Ep,q

r ! Ep+r,q!r+1 is just induced by ourdifferential d. Notice that d sends r-closed (p, q)-strips Sp,q

r to (p + r, q ! r + 1)-strips Sp+r,q!r+1, by the definition “r-closed”. By Exercise 2.7.G(b), the image lies

in Sp+r,q!r+1r .

2.7.I. EXERCISE. Verify that d sends

dSp!(r!1),q+(r!1)!1r!1 +Sp+1,q!1

r!1 ! dS(p+r)!(r!1),(q!r+1)+(r!1)!1r!1 +S

(p+r)+1,(q!r+1)!1r!1 .

(The first term on the left goes to 0 from d2 = 0, and the second term on the leftgoes to the first term on the right.)

August 26, 2010. 53

Thus we may define

dr : Ep,qr =

Sp,qr

dSp!(r!1),q+(r!1)!1r!1 + Sp+1,q!1

r!1

!

Sp+r,q!r+1r

dSp+1,q!1r!1 + Sp+r+1,q!r

r!1

= Ep+r,q!r+1r

and clearly d2r = 0 (as we may interpret it as taking an element of Sp,q

r and apply-ing d twice).

We have accomplished Goal 2.7.8(c).

2.7.12. Verifying that the cohomology of dr at Ep,qr is Ep,q

r+1. We are left with theunpleasant job of verifying that the cohomology of

(2.7.12.1)Sp!r,q+r!1

r

dSp!2r+1,q!3r!1 +Sp!r+1,q+r!2

r!1

dr $$ Sp,qr

dSp!r+1,q+r!2r!1 +Sp+1,q!1

r!1

dr $$ Sp+r,q!r+1r

dSp+1,q!1r!1 +Sp+r+1,q!r

r!1

is naturally identified with

Sp,qr+1

dSp!r,q+r!1r + Sp+1,q!1

r

and this will conclude our final Goal 2.7.8(d).We begin by understanding the kernel of the right map of (2.7.12.1). Suppose

a " Sp,qr is mapped to 0. This means that da = db + c, where b " Sp+1,q!1

r!1 .

If u = a ! b, then u " Sp,q, while du = c " Sp+r+1,q!rr!1 + Sp+r+1,q!r, from

which u is r-closed, i.e. u " Sp,qr+1. Hence a = b + u + x where dx = 0, from

which a ! x = b + c " Sp+1,q!1r!1 + Sp,q

r+1. However, x " Sp,q, so x " Sp,qr+1 by

Exercise 2.7.G(b). Thus a " Sp+1,q!1r!1 +Sp,q

r+1. Conversely, any a " Sp+1,q!1r!1 +Sp,q

r+1

satisfiesda " dSp+r,q!r+1

r!1 + dSp,qr+1 + dSp+r,q!r+1

r!1 + Sp+r+1,q!rr!1

(using dSp,qr+1 + Sp+r+1,q!r

0 and Exercise 2.7.G(b)) so any such a is indeed in thekernel of

Sp,qr !

Sp+r,q!r+1r

dSp+1,q!1r!1 + Sp+r+1,q!r

r!1

.

Hence the kernel of the right map of (2.7.12.1) is

ker =Sp+1,q!1

r!1 + Sp,qr+1

dSp!r+1,q+r!2r!1 + Sp+1,q!1

r!1

.

Next, the image of the left map of (2.7.12.1) is immediately

im =dSp!r,q+r!1

r + dSp!r+1,q+r!2r!1 + Sp+1,q!1

r!1

dSp!r+1,q+r!2r!1 + Sp+1,q!1

r!1

=dSp!r,q+r!1

r + Sp+1,q!1r!1

dSp!r+1,q+r!2r!1 + Sp+1,q!1

r!1

(as Sp!r,q!r+1r contains Sp!r+1,q+r!1

r!1 ).


Thus the cohomology of (2.7.12.1) is

ker / im =Sp+1,q!1

r!1 + Sp,qr+1

dSp!r,q+r!1r + Sp+1,q!1

r!1

=Sp,q

r+1

Sp,qr+1 / (dSp!r,q+r!1

r + Sp+1,q!1r!1 )

where the equality on the right uses the fact that dSp!r,q+r+1r + Sp,q

r+1 and anisomorphism theorem. We thus must show


r + Sp+1,q!1r!1 ) = dSp!r,q+r!1

r + Sp+1,q!1r .

However,


r + Sp+1,q!1r!1 ) = dSp!r,q+r!1

r + Sp,qr+1 / Sp+1,q!1

r!1

and Sp,qr+1 / Sp+1,q!1

r!1 consists of (p, q)-strips whose differential vanishes up to row

p + r, from which Sp,qr+1 / Sp+1,q!1

r!1 = Sp,qr as desired.

This completes the explanation of how spectral sequences work for a first-quadrant double complex. The argument applies without significant change tomore general situations, including filtered complexes.

CHAPTER 3

Sheaves

It is perhaps suprising that geometric spaces are often best understood interms of (nice) functions on them. For example, a differentiable manifold thatis a subset of Rn can be studied in terms of its differentiable functions. Because“geometric spaces” can have few (everywhere-defined) functions, a more preciseversion of this insight is that the structure of the space can be well understoodby considering all functions on all open subsets of the space. This informationis encoded in something called a sheaf. Sheaves were introduced by Leray in the1940’s. (The reason for the name is will be somewhat explained in Remark 3.4.2.)We will define sheaves and describe useful facts about them. We will begin with amotivating example to convince you that the notion is not so foreign.

One reason sheaves are slippery to work with is that they keep track of a hugeamount of information, and there are some subtle local-to-global issues. There arealso three different ways of getting a hold of them.

• in terms of open sets (the definition §3.2) — intuitive but in some way theleast helpful

• in terms of stalks (see §3.4)• in terms of a base of a topology (§3.7).

Knowing which to use requires experience, so it is essential to do a number ofexercises on different aspects of sheaves in order to truly understand the concept.

3.1 Motivating example: The sheaf of differentiable functions.

Consider differentiable functions on the topological space X = Rn (or moregenerally on a smooth manifold X). The sheaf of differentiable functions on X isthe data of all differentiable functions on all open subsets on X. We will see howto manage this data, and observe some of its properties. On each open set U + X,we have a ring of differentiable functions. We denote this ring of functions O(U).

Given a differentiable function on an open set, you can restrict it to a smalleropen set, obtaining a differentiable function there. In other words, if U + V is aninclusion of open sets, we have a “restriction map” resV,U : O(V) ! O(U).

Take a differentiable function on a big open set, and restrict it to a mediumopen set, and then restrict that to a small open set. The result is the same as if yourestrict the differentiable function on the big open set directly to the small open set.

55


In other words, if U "! V "! W, then the following diagram commutes:

O(W)resW,V $$

resW,U ''4444

4444

4O(V)

resV,U555555

5555

5

O(U)

Next take two differentiable functions f1 and f2 on a big open set U, and anopen cover of U by some {Ui}. Suppose that f1 and f2 agree on each of these Ui.Then they must have been the same function to begin with. In other words, if{Ui}i%I is a cover of U, and f1, f2 " O(U), and resU,Ui

f1 = resU,Uif2, then f1 = f2.

Thus we can identify functions on an open set by looking at them on a covering bysmall open sets.

Finally, given the same U and cover {Ui}, take a differentiable function oneach of the Ui — a function f1 on U1, a function f2 on U2, and so on — and theyagree on the pairwise overlaps. Then they can be “glued together” to make onedifferentiable function on all of U. In other words, given fi " O(Ui) for all i, suchthat resUi,Ui(Uj

fi = resUj,Ui(Ujfj for all i and j, then there is some f " O(U)

such that resU,Uif = fi for all i.

The entire example above would have worked just as well with continuousfunction, or smooth functions, or just plain functions. Thus all of these classesof “nice” functions share some common properties. We will soon formalize theseproperties in the notion of a sheaf.

3.1.1. The germ of a differentiable function. Before we do, we first give anotherdefinition, that of the germ of a differentiable function at a point p " X. Intuitively,it is a “shred” of a differentiable function at p. Germs are objects of the form{(f, open U) : p " U, f " O(U)} modulo the relation that (f,U) ! (g, V) if there issome open set W + U,V containing p where f|W = g|W (i.e., resU,W f = resV,W g).In other words, two functions that are the same in a neighborhood of p (but maydiffer elsewhere) have the same germ. We call this set of germs the stalk at p, anddenote it Op. Notice that the stalk is a ring: you can add two germs, and getanother germ: if you have a function f defined on U, and a function g defined onV , then f + g is defined on U / V . Moreover, f + g is well-defined: if f ! has thesame germ as f, meaning that there is some open set W containing p on whichthey agree, and g ! has the same germ as g, meaning they agree on some open W !

containing p, then f ! + g ! is the same function as f + g on U / V / W / W !.Notice also that if p " U, you get a map O(U) ! Ox. Experts may already see

that we are talking about germs as colimits.We can see that Op is a local ring as follows. Consider those germs vanishing

at p, which we denote mp + Op. They certainly form an ideal: mp is closed underaddition, and when you multiply something vanishing at p by any other function,the result also vanishes at p. We check that this ideal is maximal by showing thatthe quotient map is a field:

(3.1.1.1) 0 $$ m := ideal of germs vanishing at p $$ Opf)!f(p)$$ R $$ 0

3.1.A. EXERCISE. Show that this is the only maximal ideal of Op. (Hint: show thatevery element of Op \ m is invertible.)

August 26, 2010. 57

Note that we can interpret the value of a function at a point, or the value ofa germ at a point, as an element of the local ring modulo the maximal ideal. (Wewill see that this doesn’t work for more general sheaves, but does work for thingsbehaving like sheaves of functions. This will be formalized in the notion of a local-ringed space, which we will see, briefly, in §7.3.

3.1.2. Aside. Notice that m/m2 is a module over Op/m != R, i.e. it is a real vectorspace. It turns out to be naturally (whatever that means) the cotangent space tothe manifold at p. This insight will prove handy later, when we define tangent andcotangent spaces of schemes.

3.1.B. EXERCISE FOR THOSE WITH DIFFERENTIAL GEOMETRIC BACKGROUND. Provethis.

3.2 Definition of sheaf and presheaf

We now formalize these notions, by defining presheaves and sheaves. Presheavesare simpler to define, and notions such as kernel and cokernel are straightforward.Sheaves are more complicated to define, and some notions such as cokernel re-quire more thought. But sheaves are more useful because they are in some vaguesense more geometric; you can get information about a sheaf locally.

3.2.1. Definition of sheaf and presheaf on a topological space X.To be concrete, we will define sheaves of sets. However, Sets can be replaced

by any category, and other important examples are abelian groups Ab, k-vectorspaces Veck, rings Rings, modules over a ring ModA, and more. (You may haveto think more when dealing with a category of objects that aren’t “sets with addi-tional structure”, but there aren’t any new complications. In any case, this won’tbe relevant for us. ) Sheaves (and presheaves) are often written in calligraphicfont, or with an underline. The fact that F is a sheaf on a topological space X isoften written as

F

X

3.2.2. Definition: Presheaf. A presheaf F on a topological space X is the follow-ing data.

• To each open set U + X, we have a set F(U) (e.g. the set of differentiablefunctions in our motivating example). (Notational warning: Several notations arein use, for various good reasons: F(U) = -(U,F) = H0(U,F). We will use themall.) The elements of F(U) are called sections of F over U.

• For each inclusion U "! V of open sets, we have a restriction map resV,U :F(V) ! F(U) (just as we did for differentiable functions).

The data is required to satisfy the following two conditions.• The map resU,U is the identity: resU,U = idF(U).

Index

0-ring, 11A-scheme, 126A[[x]], 96A1 , 79, 80A1

Q bold, 82

A1R bold, 81

A2k , 82

An , 83An

X , 327FF(X), 123Gm blackboard bold, 154I(S), 98L mathcal with bars, 318N mathfrak, 88O(a,b) oh, 320P1

k , 111Pn

A , 112Pn

X , 329Pn

k , 112V(S), 90Aut(·), 18*•(F) cal, 308Mor, 17+i

X/Y, 418

Pic X, 272Prin X, 296Proj underline, 328Spec Z bold, 145Spec A, 79Spec Z bold, 80, 108Spec underline, 325Supp F mathcal, 72", 320AbX , Ab

preX , 62

ModOX, Mod

preOX

, 62

SetsX , SetspreX , 62

), ,, 214L(D), 297O(D), 294Ov , 258-X/k , 483!, 35", 24

#I, 91

M, 103$A , 191d-tuple embedding, 319, 320, 363–365, 377,

384, 425d-uple embedding, 184f!1 , 70f!1 , inverse image sheaf, 70f!, 60hi , 347mathcalOX,p , 78n-plane, 183pa , 360etale, 473, 475etale topology, 278, 474

abelian category, 17, 35acyclic, 439additive category, 35additive functor, 35adeles, 285adjoint, 32, 308adjoint pair, 32adjoint functors, 32adjugate matrix, 161adjunction formula, 368, 428affine cone, 186affine line, 80affine communication lemma, 125affine cone, 185, 186affine line, 79affine line with doubled origin, 110affine morphism, 164affine morphisms as Spec underline, 327affine morphisms separated, 219affine open,

26.6.F. 106.affine plane, 82affine scheme, 79, 101affine space, 83affine topology/category, 275affine variety, 126affine-local, 125

495

496 Foundations of Algebraic Geometry

Algebraic Hartogs’ Lemma, 109Algebraic Hartogs’ Lemma, 112, 128algebraic space, 461Andre-Quillen homology, 405arithmetic genus, 360arrow, 17Artinian, 243ascending chain condition, 95associated point, 133associated prime, 133assumptions on graded rings, 116automorphism, 18axiom of choice, 88axiom of choice, 11

base, 143, 200base scheme, 143, 318base change, 200base change diagram, 200base locus, 318base of a topology, 72base scheme, 200, 318base-point, 318base-point-free, 318Bertini’s theorem, 425birational, 149birational (rational) map, 149blow up, 339blow-up, 199blowing up, 339boundary, 37branch divisor, 422branch locus, 418branch point, 382

Calabi-Yau varieties, 429Cancellation Theorem for morphisms, 221canonical curve, 388canonical embedding, 388Cartesian diagram, 200Cartesian diagram/square, 26category, 17category of open sets on X, 58category of ringed spaces, 140Cech cohomology, 351Cech complex, 351change of base, 200Chevalley’s theorem, 172Chevalley’s Theorem, 172Chinese Remainder Theorem, 113class group, 272, 296closed map, 225closed point, 93, 121closed immersion, 177closed immersion affine-local, 177closed morphism, 167, 174closed subscheme, 177closed subscheme exact sequence, 283cocycle condition for transition functions, 270

codimension, 232cofibered product, 192Cohen-Macaulay, 261, 364Cohen-Seidenberg Going-Up theorem, 161coherent sheaf, 283, 285cohomology of a double complex, 44cokernel, 36colimit, f30complete linear system, 318complete (k-scheme), 226complete linear system, 318completion, 264complex, 37cone over quadric surface, 235cone over smooth quadric surface, 113conic, 182connected, 97, 121connected component, 97connecting homomorphism, 348conormal sheaf, 408constant presheaf, 60constructable set, 172constructable subset of a Noetherian scheme,

171convergence of spectral sequence, 46coproduct, 28, 30coproduct of schemes, 461coproduct of schemes, 191cotangent sheaf, 401, 411cotangent complex, 406cotangent space, 247cotangent vector, 401cotangent vector = differential, 247counit of adjunction, 33covariant, 19covering space, 473Cremona transformation, 322Cremona transformation, 152cubic, 182curve, 232cusp, 208, 253, 345cycle, 37

Dedekind domain, 209, 260degenerate, 319degree of line bundle on curve, 359degree of a point, 127degree of a projective scheme, 363degree of a rational map, 150degree of a finite morphism, 288degree of a projective scheme, 363degree of coherent sheaf on curve, 361degree of divisor on projective curve, 359degree of invertible sheaf on Pn

k , 297derivation, 409derived category, 490derived functor, 437derived functor cohomology, 347

Bibliography 497

descending chain condition, 95, 243descent, 198desingularization, 338determinant, 172determinant bundle, 282diagonal morphism ), 213diagonalizing quadrics, 130differential = cotangent vector, 247dimension, 231dimensional vanishing of cohomology, 348direct limit, 30direct image sheaf, 60discrete topology, 66discrete valuation, 258discrete valuation ring, 258disjoint union (of schemes), 107distinguished affine base, 275distinguished open set, 85, 92divisor of zeros and poles, 294domain of definition of rational map, 223dominant, 149dominant rational map, 149double complex, 43dual numbers, 88dual numbers, 81dual of a locally free sheaf, 271dual of an OX -module, 63dual projective space, 425dual projective bundle, 425dual variety, 427dualizing complex, 490dualizing sheaf, 349, 483DVR, 258

effective Cartier divisor, 179, 300effective Weil divisor, 293elliptic curve, 272elliptic curve, 390embedded points, 132embedding, 473enough injectives, 437enough projectives, 437epi morphism, 36epimorphism, 27equidimensional, 231essentially surjective, 22Euler characteristic, 293, 358Euler exact sequence, 415Euler test, 253exact, 37exceptional divisor, 199exceptional divisor, 339exponential exact sequence, 68Ext functors, 437extending the base field, 198extension by zero, 72, 314extension of an ideal, 197exterior algebra, 281

factorial, 129, 296, 297faithful functor, 20faithfully flat, 452faithfully flat, 452Faltings’ Theorem (Mordell’s Conjecture), 385fiber above a point, 200fiber diagram, 200fibered diagram/square, 26fibered product of schemes, 191final object, 25finite implies projective, 330finite presentation, 170finite extension of rings, 168finite module, 166finite morphism is closed, 174finite morphism is quasifinite, 169finite morphisms are affine, 166finite morphisms are projective, 330finite morphisms separated, 219finite presentation, 283finite type, 169finite type A-scheme, 126finite type (quasicoherent) sheaf, 285finitely generated, 283finitely generated graded ring (over A), 116finitely generated modules over discrete

valuation rings, 287finitely presented module, 40finitely presented algebra, 403flabby sheaf, 443flasque sheaf, 443flat, 41, 335, 434, 447flat limit, 456flat A-module, 448flat lft morphisms are open, 470flat morphism, 450flat of relative dimension n, 452flat quasicoherent sheaf, 450flat quasicoherent sheaf over a base, 450flat ring homomorphism, 448flex line, 394forgetful functor, 20formally etale, 482formally smooth, 482formally unramified, 482fractional linear transformations, 318free sheaf, 269, 270Freyd-Mitchell Embedding Theorem, 36Frobenius morphism, 375full functor, 20function field K(·), 123function field, 123, 132, 235functions on a scheme, 79, 106functor, 19functor of points, 145

Gaussian integers mathbbZ[i], 260Gaussian integers mathbbZ[i], 254


generalization, 94generated by global sections, 306generated in degree 1, 116, 183generic point, 121generic fiber, 200generic point, 94generically separable morphism, 421generization, 121geometric fiber, 204geometric fiber, 203geometric point, 204geometrically connected, 204geometrically

connected/irreducible/integral/reducedfibers, 203

geometrically integral, 204geometrically irreducible, 204geometrically nonsingular fibers, 475geometrically reduced, 204germ, 58germ of function near a point, 107globally generated, 306gluability axiom, 58gluing along closed subschemes, 461Going-Up theorem, 161graded ring, 116graded ring over A, 116graph morphism, 220graph of rational map, 190Grassmannian, 119, 157, 416Grothendieck topology, 474Grothendieck topology, 278group scheme, 155group schemes, 154groupoid, 18

Hartogs’ Theorem, 272Hausdorff, 213, 213, 215height, 232higher direct image sheaf, 368, 369higher pushforward sheaf, 369Hilbert polynomial, 362Hilbert basis theorem, 95Hilbert function, 362Hilbert scheme, 460Hironaka’s example, 462Hodge bundle, 464Hodge theory, 420Hom, 35homogeneous ideal, 115homogeneous space, 480homogeneous ideal, 116homology, 37homotopic maps of complexes, 434Hopf algebra, 156hypercohomology, 44hyperplane, 182, 183hyperplane class, 296

hypersurface, 182, 233

ideal denominators, 242ideal of denominators, 129ideal sheaf, 177identity axiom, 58immersion, 473index category, 28induced reduced subscheme structure, 189induced reduced subscheme structure, 189infinite-dimensional Noetherian ring, 233initial object, 25injective limit, 30injective object in an abelian category, 437integral, 123, 160integral closure, 207integral extension of rings, 160integral morphism, 169integral morphism of rings, 160intersection number, 367inverse image, 71inverse image ideal sheaf, 200inverse image scheme, 200inverse image sheaf, 70inverse limit, 29invertible ideal sheaf, 179invertible sheaf, 270, 273irreducible, 93, 121irreducible (Weil) divisor, 293irreducible component, 96irreducible components, 121irregularity, 421irrelevant ideal, 116isomorphism, 18isomorphism of schemes, 106

Jacobian, 404Jacobian matrix, 476Jacobian criterion, 250Jacobson radical, 163

K3 surfaces, 429kernel, 36knotted plane, 260Kodaira vanishing, 349Krull, 239Krull dimension, 231Krull dimension, 231Krull’s Principal Ideal Theorem, 239

Luroth’s theorem, 424left-adjoint, 32left-exact, 40left-exactness of global section functor, 70Leibniz rule, 402length, 367Leray spectral sequence, 348limit, 29line, 183

Bibliography 499

line bundle, 269linear space, 182linear series, 318linear system, 318local complete intersection, 428local criterion for flatness, 457local ringed space, 107local-ringed space, 142locally ringed spaces, 142locally closed immersion, 180locally constant sheaf, 60locally free sheaf, 270locally free sheaf, 269, 273locally integral (temp.), 254locally Noetherian scheme, 126locally of finite type A-scheme, 126locally of finite presentation, 170locally of finite type, 169locally principal subscheme, 179locally principal Weil divisor, 295long exact sequence, 39long exact sequence of higher pushforward

sheaves, 369

magic diagram, 27mapping cone, 50, 354minimal prime, 93, 96module of Kahler differentials, 402module of relative differentials, 402moduli space, 384, 392monic morphism, 36monomorphism, 27Mordell’s conjecture, 385morphism, 17morphism of (pre)sheaves, 62morphism of (pre)sheaves, 62morphism of ringed spaces, 140morphism of ringed spaces, 140morphism of schemes, 143multiplicity of a singularity, 346

Nagata, 233, 299Nagata’s Lemma, 299Nakayama’s Lemma, 162, 163, 173nilpotents, 88, 122nilradical, 88, 88, 91node, 208, 253Noetherian induction, 97Noetherian ring, 95, 95Noetherian rings, important facts about, 95Noetherian scheme, 121, 126Noetherian topological space, 93, 95non-archimedean, 258non-archimedean analytic geometry, 278, 288non-degenerate, 319non-zero-divisor, 23nonsingular, 247, 251nonsingularity, 247normal, 109, 128

normal = R1+S2, 260normal exact sequence, 428normal sheaf, 408normalization, 206Nullstellensatz, 83, 127number field, 209

object, 17octic, 182Oka’s theorem, 285, 288open immersion of ringed spaces, 140open subscheme, 106open immersion, 159, 159open subscheme, 159opposite category, 20orientation of spectral sequence, 44

page of spectral sequence, 44partially ordered set, 19partition of unity, 353Picard group, 272plane, 183points, A-valued, 145points, S-valued, 145pole, 259poset, 19presheaf, 57presheaf cokernel, 63presheaf kernel, 63primary ideal, 134prime avoidance (temp. notation), 239principal divisor, 296principal Weil divisor, 295product, 22, 191Proj, 116projection formula, 370projective coordinates, 115projective space, 112projective A-scheme, 116projective X-scheme, 329projective and quasifinite implies finite, 331projective cone, 186projective coordinates, 112projective distinguished open set, 117projective line, 111projective module, 436projective morphism, 329projective object in an abelian category, 436projective space, 118projective variety, 126projectivization of a locally free sheaf, 329proper, 226proper non-projective surface, 460proper transform, 338, 339Puisseux series, 258pullback diagram, 200pullback for [locally?] ringed spaces, 316pure dimension, 231pushforward sheaf, 60


pushforward of coherent sheaves, 371pushforward of quasicoherent sheaves, 311pushforward sheaf, 60

quadric, 182quadric surface, 235quadric surface, 185quartic, 182quasicoherent sheaf, 269quasicoherent sheaf, 103, 273quasicoherent sheaves: product, direct sum,

#, Sym, cokernel, image, ", 281quasicompact, 121quasicompact morphism, 164quasicompact topological space, 93quasifinite, 170quasiisomorphism, 352quasiprojective morphism, 355quasiprojective scheme, 119quasiprojective is separated, 219quasiseparated, 218quasiseparated morphism, 164quasiseparated scheme, 121quintic, 182quotient object, 36quotient sheaf, 67

radical, 90radical ideal, 87radical ideal, 91ramification point, 382ramification divisor, 422ramification locus, 418rank of locally free sheaf, 273rank of coherent sheaf on curve, 361rank of finite type quasicoherent sheaf, 287rank of quadratic, 130rational map, 148rational function, 132rational normal curve, 319rational normal curve, 184rational normal curve take 1, 94rational section of invertible sheaf, 272reduced, 123, 126reduced ring, 88reduced scheme, 122reducedness is stalk-local, 123reduction, 190Rees algebra, 338reflexive sheaf, 408regular, 247regular scheme, 251regular function, 132regular local ring, 251regular point, 247regular section of invertible sheaf, 272relative (co)tangent sheaf, 411relative (co)tangent vectors, 401representable functor, 153

residue field, 106residue field at a point, 107Residue theorem, 335, 359resolution of singularities, 338restriction map, 57restriction of a quasicoherent sheaf, 312restriction of sheaf to open set, 63resultant, 172Riemann-Roch for coherent sheaves on a

curve, 361Riemann-Roch for surfaces, 368right exact, 24right-adjoint, 32right-exact, 40ring scheme, 155ring of integers in a number field, 209ring scheme, 156ringed space, 61, 77rulings on the quadric surface, 185

S2, 260Sard’s theorem, 479saturated module, 309saturation map, 308scheme over A, 126scheme, definition of, 106scheme-theoretic inverse image, 200scheme-theoretic pullback, 200Schubert cell, 157sections over an open set, 57Segre embedding, 205, 320Segre product, 205Segre variety, 205separable morphism, 421separated, 110, 215separated over A, 215separated presheaf, 58separatedness, 106septic, 182Serre duality, 349Serre duality (strong form), 484Serre vanishing, 349Serre’s criterion for normality, 260Serre’s criterion for affineness, 369sextic, 182sheaf, 57sheaf Hom (Hom underline), 63sheaf Hom (Hom underline) of quasicoherent

sheaves, 285sheaf Hom (underline), 63sheaf determined by sheaf on base, 276sheaf of ideals, 177sheaf of relative differentials, 401sheaf on a base, 72, 73sheaf on a base determines sheaf, 73sheaf on affine base, 275sheafification, 64, 66singular, 247, 251

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site, 278skyscraper sheaf, 59smooth, 247, 473, 475smooth quadric surface, 130specialization, 94, 121spectral sequence, 43spectrum, 79stack, 74, 278stalk, 58stalk-local, 123, 125strict transform, 339strong Serre duality, 484structure morphism, 144structure sheaf, 77structure sheaf (of ringed space), 61structure sheaf on Spec A, 101submersion, 473subobject, 36subscheme cut out by a section of a locally

free sheaf, 272subsheaf, 67support, 286support of a sheaf, 72support of a Weil divisor, 293surface, 232surjective morphism, 202symbolic power of an ideal, 244symmetric algebra, 281

tacnode, 208, 253tame ramification, 423tangent line, 394tangent sheaf, 411tangent space, 247tangent vector, 247tautological bundle, 327tensor algebra T!

A(M), 281tensor product, 23, 24tensor product of O-modules, 70tensor product of sheaves, 70topos, 278torsion-free, 282total fraction ring, 132total space of locally free sheaf, 327total transform, 339trace map, 483transcendence basis/degree, 234transition functions, 270transitive group action, 480trigonal curve, 387twisted cubic, 182twisted cubic, 235twisted cubic curve, 94twisting by a line bundle, 305two-plane example, 261

ultrafilter, 98underline S, 60underline Spre, 60

uniformizer, 256unit of adjunction, 33universal property, 15universal property of blowing up, 339universally, 225universally closed, 225unramified, 473, 475uppersemicontinuity of rank of finite type

sheaf, 287

valuation, 258valuation ring, 258valuative criterion for separatedness, 261value of a function, 79value of a quasicoherent sheaf at a point, 287value of function, 106value of function at a point, 107vanishing set, 90vanishing theorems, 358vanishing scheme, 179variety, 213, 217vector bundle, 327Veronese, 319Veronese embedding, 320Veronese subring, 147Veronese embedding, 184, 319, 363–365, 377,

384, 425Veronese surface, 184vertical (co)tangent vectors, 401

Weierstrass normal form, 395weighted projective space, 185Weil divisor, 293wild ramification, 423

Yoneda’s lemma, 195Yoneda’s lemma, 28

Zariski (co)tangent space, 247Zariski tangent space, 247Zariski topology, 90, 90zero ring, 11zero object, 25, 35zero-divisor, 23

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